00footnotetext: 2000 Mathematics Subject Classification: 57N05, 20F38,
20F0500footnotetext: Keywords: Mapping class groups, nonorientable surfaces,
twist subgroup, torsion, generating sets

# Torsion Generators of the Twist Subgroup

Tüli̇n Altunöz, Mehmetci̇k Pamuk, and Oğuz Yıldız  Department of Mathematics,
Middle East Technical University, Ankara, Turkey<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

We showed that the twist subgroup of the mapping class group of a closed
connected nonorientable surface of genus $g\geq 13$ can be generated by two
involutions and an element of order $g$ or $g-1$ depending on whether $g$ is
odd or even respectively.

##  1\. Introduction

Let $\Sigma_{g}$ denote a closed connected orientable surface of genus $g$.
The mapping class group, ${\rm Mod}(\Sigma_{g})$, is the group of the isotopy
classes of orientation preserving diffeomorphisms of $\Sigma_{g}$. It is a
classical result that ${\rm Mod}(\Sigma_{g})$ is generated by finitely many
Dehn twists about nonseparating simple closed curves [4, 6, 12]. The study of
algebraic properties of mapping class group, finding small generating sets,
generating sets with particular properties, is an active one leading to
interesting developments. Wajnryb [23] showed that ${\rm Mod}(\Sigma_{g})$ can
be generated by two elements given as a product of Dehn twists. As the group
is not abelian, this is the smallest possible. Later, Korkmaz [9] showed that
one of these generators can be taken as a Dehn twist, he also proved that
${\rm Mod}(\Sigma_{g})$ can be generated by two torsion elements. Recently,
the third author showed that ${\rm Mod}(\Sigma_{g})$ is generated by two
torsions of small orders [24].

Generating ${\rm Mod}(\Sigma_{g})$ by involutions was first considered by
McCarthy and Papadopoulus [15]. They showed that the group can be generated by
infinitely many conjugates of a single involution (element of order two) for
$g\geq 3$. In terms of generating by finitely many involutions, Luo [14]
showed that any Dehn twist about a nonseparating simple closed curve can be
written as a product six involutions, which in turn implies that ${\rm
Mod}(\Sigma_{g})$ can be generated by $12g+6$ involutions. Brendle and Farb
[2] obtained a generating set of six involutions for $g\geq 3$. Following
their work, Kassabov [7] showed that ${\rm Mod}(\Sigma_{g})$ can be generated
by four involutions if $g\geq 7$. Recently, Korkmaz [10] showed that ${\rm
Mod}(\Sigma_{g})$ is generated by three involutions if $g\geq 8$ and four
involutions if $g\geq 3$. Also, the third author improved his result showing
that it is generated by three involutions if $g\geq 6$ [25].

The main aim of this paper is to find minimal generating sets of torsion
elements for a particular subgroup, namely the twist subgroup, of the mapping
class groups of nonorientable surfaces. Let $N_{g}$ denote a closed connected
nonorientable surface of genus $g$. The mapping class group,${\rm
Mod}(N_{g})$, is defined to be the group of the isotopy classes of all
diffeomorphisms of $N_{g}$. Compared to orientable surfaces less is known
about ${\rm Mod}(N_{g})$. Lickorish [11, 13] showed that it is generated by
Dehn twists about two-sided simple closed curves and a so-called
$Y$-homeomorphism (or a crosscap slide). Chillingworth [3] gave a finite
generating set for ${\rm Mod}(N_{g})$ that linearly depends on $g$.
Szepietowski [21] proved that ${\rm Mod}(N_{g})$ is generated by three
elements and by four involutions.

The twist subgroup $\mathcal{T}_{g}$ of ${\rm Mod}(N_{g})$ is the group
generated by Dehn twists about two-sided simple closed curves. The group
$\mathcal{T}_{g}$ is a subgroup of index $2$ in ${\rm Mod}(N_{g})$ [13].
Chillingworth [3] showed that $\mathcal{T}_{g}$ can be generated by finitely
many Dehn twists. Stukow [19] obtained a finite presentation for
$\mathcal{T}_{g}$ with $(g+2)$ Dehn twist generators. Later Omori [17] reduced
the number of Dehn twist generators to $(g+1)$ for $g\geq 4$. If it is not
required that all generators are Dehn twists, Du [5] obtained a generating set
consisting of three elements, two involutions and an element of order $2g$
whenever $g\geq 5$ and odd.

In the present paper, we prove that $\mathcal{T}_{g}$ can be generated by two
involutions and an element of order $g$ or $g-1$ depending on the parity of
$g$ (see Theorems 2.4 and 2.5).

######  Main Theorem.

The twist subgroup $\mathcal{T}_{g}$ can be generated by two involutions and
an element of order $g$ or $g-1$ depending on whether $g$ is odd or even
respectively.

Before we finish the introduction, let us point out that the twist subgroup
$\mathcal{T}_{g}$ admits an epimorphism onto the automorphism group of
$H_{1}(N_{g};\mathbb{Z}_{2})$ preserving the $\pmod{2}$ intersection pairing
[16], which is isomorphic to (see [8] and [22])

$\displaystyle\begin{cases}Sp(2h;\mathbb{Z}_{2})&\text{if $g=2h+1$,}\\\
Sp(2h;\mathbb{Z}_{2})\ltimes\mathbb{Z}_{2}^{2h+1}&\text{if
$g=2h+2$.}\end{cases}$

Hence, the action of mapping classes on $H_{1}(N_{g};\mathbb{Z}_{2})$ induces
an epimorphism from $\mathcal{T}_{g}$ to
$Sp\big{(}2\lfloor\dfrac{g-1}{2}\rfloor;\mathbb{Z}_{2}\big{)}$, which
immediately implies the following corollary:

######  Corollary 1.1.

The symplectic group
$Sp\big{(}2\lfloor\dfrac{g-1}{2}\rfloor;\mathbb{Z}_{2}\big{)}$ can be
generated by two involutions and an element of order $g$ or $g-1$ depending on
whether $g$ is odd or even respectively.

Acknowledgments. The authors thank Mustafa Korkmaz for helpful conversations
and Tara Brendle for her helpful comments on a previous version of this paper.
The first author was partially supported by the Scientific and Technologic
Research Council of Turkey (TÜBİTAK)[grant number 117F015].

##  2\. Background and Results on Mapping Class Groups

Let $N_{g}$ be a closed connected nonorientable surface of genus $g$. Note
that the genus for a nonorientable surface is the number of projective planes
in a connected sum decomposition. The mapping class group ${\rm Mod}(N_{g})$
of the surface $N_{g}$ is defined to be the group of the isotopy classes of
diffeomorphisms $N_{g}\to N_{g}$. Throughout the paper we do not distinguish a
diffeomorphism from its isotopy class. For the composition of two
diffeomorphisms, we use the functional notation; if $g$ and $h$ are two
diffeomorphisms, the composition $gh$ means that $h$ acts on $N_{g}$ first.
A simple closed curve on a nonorientable surface $N_{g}$ is said to be one-
sided if a regular neighbourhood of it is homeomorphic to a Möbius band. It is
called two-sided if a regular neighbourhood of it is homeomorphic to an
annulus. If $a$ is a two-sided simple closed curve on $N_{g}$, to define the
Dehn twist $t_{a}$, we need to fix one of two possible orientations on a
regular neighbourhood of $a$ (as we did for the curve $a_{1}$ in Figure 1).
Following [10] the right-handed Dehn twist $t_{a}$ about $a$ will be denoted
by the corresponding capital letter $A$.

Now, let us recall the following basic properties of Dehn twists which we use
frequently in the remaining of the paper. Let $a$ and $b$ be two-sided simple
closed curves on $N_{g}$ and $f\in{\rm Mod}(N_{g})$.

  * •

Commutativity: If $a$ and $b$ are disjoint, then $AB=BA$.

  * •

Conjugation: If $f(a)=b$, then $fAf^{-1}=B^{s}$, where $s=\pm 1$ depending on
whether $f$ is orientation preserving or orientation reversing on a
neighbourhood of $a$ with respect to the chosen orientation.

Figure 1. The curves $a_{1},a_{2},b_{i},c_{i},e,f$ and $\gamma_{i}$ on the
surface $N_{g}$.

Figure 2. Generators of $H_{1}(N_{g};\mathbb{R})$.

Consider the surface $N_{g}$ shown in Figure 1. The Dehn twist generators of
Omori can be given as follows (note that we do not have the curve $d_{r}$ when
$g$ is odd).

######  Theorem 2.1.

[17] The twist subgroup $\mathcal{T}_{g}$ is generated by the following
$(g+1)$ Dehn twists

  1. (1)

$A_{1},A_{2},B_{1},\ldots,B_{r}$, $C_{1},\ldots,C_{r-1}$ and $E$ if $g=2r+1$
and

  2. (2)

$A_{1},A_{2},B_{1},\ldots,B_{r}$, $C_{1},\ldots,C_{r-1}$, $D_{r}$ and $E$ if
$g=2r+2$.

Consider a basis $\\{x_{1},x_{2}.\ldots,x_{g-1}\\}$ for
$H_{1}(N_{g};\mathbb{R})$ such that the curves $x_{i}$ are one-sided and
disjoint as in Figure 2. It is known that every diffeomorphism $f:N_{g}\to
N_{g}$ induces a linear map $f_{\ast}:H_{1}(N_{g};\mathbb{R})\to
H_{1}(N_{g};\mathbb{R})$. Therefore, one can define a homomorphism $D:{\rm
Mod}(N_{g})\to\mathbb{Z}_{2}$ by $D(f)=\textrm{det}(f_{\ast})$. The following
lemma from [11] tells when a mapping class falls into the twist subgroup
$\mathcal{T}_{g}$.

######  Lemma 2.2.

Let $f\in{\rm Mod}(N_{g})$. Then $D(f)=1$ if $f\in\mathcal{T}_{g}$ and
$D(f)=-1$ if $f\not\in\mathcal{T}_{g}$.

###  2.1. A generating set for $\mathcal{T}_{g}$

We start with presenting a generating set for $\mathcal{T}_{g}$. The
diffeomorphism $T$ is the rotation by $\frac{2\pi}{g}$ or $\frac{2\pi}{g-1}$
as shown on the right hand sides of Figures 3 and 4, respectively. Note that
the rotation $T$ satisfies $D(T)=1$, which implies that $T$ belongs to
$\mathcal{T}_{g}$.

Figure 3. The models for $N_{g}$ if $g=2r+1$.

Figure 4. The models for $N_{g}$ if $g=2r+2$.

######  Theorem 2.3.

The twist subgroup $\mathcal{T}_{g}$ is generated by the elements

  1. (1)

$T,A_{1}A_{2}^{-1},B_{1}B_{2}^{-1}$ and $E$ if $g=2r+1$ and $r\geq 3$,

  2. (2)

$T,A_{1}A_{2}^{-1},B_{1}B_{2}^{-1},D_{r}$ and $E$ if $g=2r+2$ and $r\geq 3$.

###### Proof.

Let $G$ be the subgroup of $\mathcal{T}_{g}$ generated by the following set

$G=\left\\{\begin{array}[]{lll}\\{T,A_{1}A_{2}^{-1},B_{1}B_{2}^{-1},E\\}&\textrm{if}&g=2r+1,\\\
\\{T,A_{1}A_{2}^{-1},B_{1}B_{2}^{-1},D_{r},E\\}&\textrm{if}&g=2r+2,\\\
\end{array}\right.$

where $r\geq 3$. It follows from Theorem 2.1 that we only need to prove that
$G$ contains the elements $A_{1},A_{2},B_{i}$ and $C_{j}$ shown in Figures 3
and 4 where $i=1,\ldots,r$ and $j=1,\ldots,r-1$. (We use the explicit
homeomorphism constructed in [18, Section $3$] to identify the models in these
figures.)
Let $\mathcal{S}$ denote the set of isotopy classes of two-sided non-
separating simple closed curves on $N_{g}$. Define a subset $\mathcal{G}$ of
$\mathcal{S}\times\mathcal{S}$ as

$\mathcal{G}=\\{(a,b):AB^{-1}\in G\\}.$

Using the arguments similar to the proof of [10, Theorem $5$], the set
$\mathcal{G}$ satisfies

  * •

if $(a,b)\in\mathcal{G}$, then $(b,a)\in\mathcal{G}$ (symmetry),

  * •

if $(a,b)\ \textrm{and}\ (b,c)\in\mathcal{G}$, then $(a,c)\in\mathcal{G}$
(transitivity) and

  * •

if $(a,b)\in\mathcal{G}$ and $H\in G$ then $(H(a),H(b))\in\mathcal{G}$
($G$-invariance).

Hence, $\mathcal{G}$ defines an equivalence relation on $\mathcal{S}$.
We begin by showing that $B_{i}C_{j}^{-1}$ is contained in $G$ for all $i,j$.
It will follow from the definition of $G$ and from the fact that
$T(b_{1},b_{2})=(c_{1},c_{2})$, we have $C_{1}C_{2}^{-1}$ is in $G$ (here, we
use the notation $f(a,b)$ to denote $(f(a),f(b))$). Also, by conjugating
$C_{1}C_{2}^{-1}$ with powers of $T$, one can show that the elements
$B_{i}B_{i+1}^{-1}$ and $C_{i}C_{i+1}^{-1}$ are contained in $G$. Moreover,
the subgroup $G$ contains the elements $B_{i}B_{j}^{-1}$ and $C_{i}C_{j}^{-1}$
by the transitivity. To start with, since $B_{2}B_{3}^{-1}\in G$ and it is
easy to check that

$B_{2}B_{3}^{-1}A_{2}A_{1}^{-1}(b_{2},b_{3})=(a_{2},b_{3}),$

so that $A_{2}B_{3}^{-1}$ is contained in the subgroup $G$. We have

$(A_{1}A_{2}^{-1})(A_{2}B_{3}^{-1})(B_{3}B_{2}^{-1})=A_{1}B_{2}^{-1}\in G,$

since each of the factors is contained in $G$. Hence,
$T(a_{1},b_{2})=(b_{1},c_{2})$ implies that $B_{1}C_{2}^{-1}$ is also in $G$.
Now, the subgroup $G$ contains the element

$B_{1}C_{1}^{-1}=(B_{1}C_{2}^{-1})(C_{2}C_{1}^{-1}).$

Therefore, the elements $B_{i}C_{i}^{-1}$ is contained in $G$ by conjugating
with powers of $T$ for all $i=1,\ldots,r-1$. It follows from the transitivity
that $B_{i}C_{j}^{-1}$ is in $G$. Note that, we have

  * •

$(A_{1}B_{2}^{-1})(B_{2}C_{1}^{-1})=A_{1}C_{1}^{-1}$,

  * •

$(C_{1}A_{1}^{-1})(A_{1}A_{2}^{-1})=C_{1}A_{2}^{-1}$, and

  * •

$(C_{2}C_{1}^{-1})(C_{1}A_{1}^{-1})=C_{2}A_{1}^{-1}$

from which it follows that the elements $A_{1}C_{1}^{-1}$, $C_{1}A_{2}^{-1}$
and $C_{2}A_{1}^{-1}$ belong to $G$. It can also be shown that

$(B_{2}A_{1}^{-1})(C_{1}A_{2}^{-1})(C_{2}A_{1}^{-1})(b_{2},a_{1})=(d_{1},a_{1})$

and

$(A_{1}B_{2}^{-1})(A_{1}C_{1}^{-1})(A_{1}C_{2}^{-1})(A_{1}B_{2}^{-1})(a_{2},a_{1})=(d_{2},a_{1})$

so that $G$ contains $D_{1}A_{1}^{-1}$ and $D_{2}A_{1}^{-1}$ (here, the curves
$d_{1}$ and $d_{2}$ are shown in [10, Figure $1$]). Also, we have

$(D_{2}A_{1}^{-1})(A_{1}C_{1}^{-1})=D_{2}C_{1}^{-1}\in G.$

By similar arguments as in the proof of [10, Theorem $5$], the lantern
relation implies the following identity

$A_{3}=(A_{2}C_{2}^{-1})(D_{1}A_{1}^{-1})(D_{2}C_{1}^{-1}).$

Since the subgroup $G$ contains each factor on the right hand side, the
element $A_{3}$ belongs to $G$. It follows from

$B_{3}=A_{3}(B_{3}B_{1}^{-1})A_{3}(B_{1}B_{3}^{-1})A_{3}^{-1}$

that $B_{3}$ is also contained in $G$. By conjugating $B_{3}$ with the powers
of $T$, we get $A_{1},B_{1},C_{1},\ldots B_{r-1},C_{r-1}$ and $B_{r}$ are all
contained in $G$. Moreover,

$A_{2}=(A_{2}A_{1}^{-1})A_{1}\in G.$

Therefore, we conclude that $G=\mathcal{T}_{g}$.

Figure 5. The involution $\sigma$ if $g=2r+1$.

Figure 6. The involution $\sigma$ if $g=2r+2$.

We consider the surface $N_{g}$ where $g$-crosscaps are distributed on the
sphere as in Figures 5 and 6. If $g=2r+1\geq 13$, there is a reflection,
$\sigma$, of the surface $N_{g}$ in the $xy$-plane such that

  * •

$\sigma(a_{1})=f$, $\sigma(b_{r})=c_{r-1}$,

  * •

$\sigma(x_{2})=x_{3}$, $\sigma(x_{g})=x_{g-2}$ and

  * •

$\sigma(x_{i})=x_{i}$ if $i=4,\ldots,g-3$ or $i=1,g-1$.

with reverse orientation as in Figure 5. (Recall that $x_{i}$’s are the
generators of $H_{1}(N_{g};\mathbb{R})$ as shown in Figure 2.) If $g=2r+2\geq
14$, there is a reflection, $\sigma$, of the surface $N_{g}$ in the $xy$-plane
such that

  * •

$\sigma(a_{1})=f$, $\sigma(b_{r})=d_{r}$,

  * •

$\sigma(x_{2})=x_{3}$, $\sigma(x_{3})=x_{4}$, $\sigma(x_{g})=x_{g-2}$ and

  * •

$\sigma(x_{i})=x_{i}$ if $i=6,\ldots,g-3$ or $i=1,g-1$.

with reverse orientation as in Figure 6. Note that in both cases the
reflection $\sigma$ in in $\mathcal{T}_{g}$ since $D(\sigma)=1$ for $g\geq
13$.

Now, for the remaining part of the paper, let $\Gamma_{i}$ denote the right
handed Dehn twist about the curve $\gamma_{i}$ shown in Figure 1.

######  Theorem 2.4.

For odd $g\geq 13$, the twist subgroup $\mathcal{T}_{g}$ is generated by the
three elements $T,\sigma$ and $\sigma\Gamma_{g-3}C_{\frac{g-9}{2}}^{-1}$.

###### Proof.

Consider the surface $N_{g}$ as in Figure 5. Since

$\sigma(\gamma_{g-3})=\gamma_{g-3}\textrm{ and
}\sigma(c_{\frac{g-9}{2}})=c_{\frac{g-9}{2}},$

and $\sigma$ reverses the orientation of a neighbourhood of a two-sided simple
closed curve, we have

$\sigma\Gamma_{g-3}\sigma=\Gamma_{g-3}^{-1}\textrm{ and }\sigma
C_{\frac{g-9}{2}}\sigma=C_{\frac{g-9}{2}}^{-1}.$

Hence, it is easy to verify that $\sigma\Gamma_{g-3}C_{\frac{g-9}{2}}^{-1}$ is
an involution. Let $H$ be the subgroup of $\mathcal{T}_{g}$ generated by the
following set

$\\{T,\sigma,\sigma\Gamma_{g-3}C_{\frac{g-9}{2}}^{-1}\\},$

where $g\geq 13$ and odd. It follows from Theorem 2.3 that we only need to
prove that the elements $A_{1}A_{2}^{-1}$, $B_{1}B_{2}^{-1}$ and $E$ are
contained in the subgroup $H$.
Since

$\Gamma_{g-3}C_{\frac{g-9}{2}}^{-1}=(\sigma)(\sigma\Gamma_{g-3}C_{\frac{g-9}{2}}^{-1}),$

the element $\Gamma_{g-3}C_{\frac{g-9}{2}}^{-1}$ belongs to $H$.
It follows from

  * •

$T^{13-g}(\gamma_{g-3},c_{\frac{g-9}{2}})=(\gamma_{10},c_{2})$,

  * •

$T^{-4}(\gamma_{10},c_{2})=(\gamma_{6},a_{1})$ and

  * •

$T^{2}(\gamma_{6},a_{1})=(\gamma_{8},c_{1})$

that the elements $\Gamma_{10}C_{2}^{-1}$, $\Gamma_{6}A_{1}^{-1}$ and
$\Gamma_{8}C_{1}^{-1}$ are in $H$. Since

$(\Gamma_{6}A_{1}^{-1})(C_{2}\Gamma_{10}^{-1})(\gamma_{6},a_{1})=(c_{2},a_{1}),$

the element $C_{2}A_{1}^{-1}$ is in $H$. Since

  * •

$(\Gamma_{6}A_{1}^{-1})(A_{1}C_{2}^{-1})=\Gamma_{6}C_{2}^{-1}\in H$ and

  * •

$T^{4}(\gamma_{6},c_{2})=(\gamma_{10},c_{4})$,

$\Gamma_{10}C_{4}^{-1}$ is contained in $H$. Also, we have

$(C_{4}\Gamma_{10}^{-1})(\Gamma_{10}C_{2}^{-1})=C_{4}C_{2}^{-1}\in H.$

Thus, $C_{3}C_{1}^{-1}$, $C_{3}C_{5}^{-1}$, $B_{4}B_{2}^{-1}$ and
$C_{2}A_{1}^{-1}$ are contained in $H$ by conjugating $C_{4}C_{2}^{-1}$ with
some powers of $T$. Then, since

$(C_{4}C_{2}^{-1})(B_{4}B_{2}^{-1})(c_{2},a_{1})=(b_{2},a_{1}),$

$H$ contains $B_{2}A_{1}^{-1}$. Also, we get

$(C_{2}A_{1}^{-1})(A_{1}B_{2}^{-1})=C_{2}B_{2}^{-1}\in H.$

Since

$C_{1}C_{3}^{-1}B_{2}B_{4}^{-1}(c_{3},c_{5})=(b_{4},c_{5}),$

$H$ contains $B_{4}C_{5}^{-1}$. Then,

$T^{-4}(b_{4},c_{5})=(b_{2},c_{3})$

implies that $B_{2}C_{3}^{-1}\in H$. It follows from the following equalities

  * •

$C_{2}C_{3}^{-1}=(C_{2}B_{2}^{-1})(B_{2}C_{3}^{-1})$ and

  * •

$T^{-2}(b_{2},c_{3})=(b_{1},b_{2})$

that $B_{1}B_{2}^{-1}$ belongs to $H$. On the other hand, since

  * •

$(\Gamma_{8}C_{1}^{-1})(C_{1}C_{3}^{-1})(C_{3}C_{5}^{-1})(C_{5}B_{4}^{-1})=\Gamma_{8}B_{4}^{-1}$
and

  * •

$T^{-7}(\gamma_{8},b_{4})=(\gamma_{1},a_{1})=(a_{2},a_{1})$

that $A_{1}A_{2}^{-1}$ is contained in $H$.
Since the elements $T$, $A_{1}A_{2}^{-1}$ and $B_{1}B_{2}^{-1}$ are contained
in the subgroup $H$, the generators
$A_{1},B_{1},C_{1},\ldots,B_{r-1},C_{r-1},B_{r}$ are in $H$ by the proof of
Theorem 2.1. Moreover,

$A_{1}\sigma(a_{1})=A_{1}(f)=e$

leads to $E\in H$. This completes the proof.

######  Theorem 2.5.

For even $g\geq 14$, the twist subgroup $\mathcal{T}_{g}$ is generated by the
three elements $T,\sigma$ and $\sigma\Gamma_{g-3}C_{\frac{g-8}{2}}^{-1}$.

###### Proof.

Consider the surface $N_{g}$ as in Figure 6. Since

$\sigma(\gamma_{g-3})=\gamma_{g-3}\textrm{ and
}\sigma(c_{\frac{g-8}{2}})=c_{\frac{g-8}{2}},$

and $\sigma$ reverses the orientation of a neighbourhood of a two-sided simple
closed curve, we have

$\sigma\Gamma_{g-3}\sigma=\Gamma_{g-3}^{-1}\textrm{ and }\sigma
C_{\frac{g-8}{2}}\sigma=C_{\frac{g-8}{2}}^{-1}.$

Hence, it is easy to show that $\sigma\Gamma_{g-3}C_{\frac{g-9}{2}}^{-1}$ is
an involution. Let $K$ be the subgroup of $\mathcal{T}_{g}$ generated by the
following set

$\\{T,\sigma,\sigma\Gamma_{g-3}C_{\frac{g-8}{2}}^{-1}\\},$

where $g\geq 14$ and even.
Since

$\Gamma_{g-3}C_{\frac{g-8}{2}}^{-1}=(\sigma)(\sigma\Gamma_{g-3}C_{\frac{g-8}{2}}^{-1}),$

the element $\Gamma_{g-3}C_{\frac{g-8}{2}}^{-1}$ is contained in $K$.
It follows from

$T^{13-g}(\gamma_{g-3},c_{\frac{g-8}{2}})=(\gamma_{10},c_{2})$

that the elements $\Gamma_{10}C_{2}^{-1}$ is in the subgroup $K$. Recall that
this element also appears in the proof of Theorem 2.4. The remaining part of
the proof follows as in the proof of the previous theorem. Also, note that the
element $D_{r}$ belongs to $K$ since $\sigma(b_{r})=d_{r}$. This finishes the
proof.

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# A broadband view on microquasar MAXI J$1820+070$ during the 2018 outburst

J. Rodi INAF - Istituto di Astrofisica e Planetologia Spaziali, via Fosso del
Cavaliere 100, 00133 Roma, Italy  A. Tramacere Department of Astronomy,
University of Geneva, Ch. d’Ecogia 16, 1290, Versoix, Switzerland  F. Onori
INAF - Istituto di Astrofisica e Planetologia Spaziali, via Fosso del
Cavaliere 100, 00133 Roma, Italy  INAF - Osservatorio Astronomico d’Abruzzo,
via M. Maggini snc, I-64100 Teramo, Italy  G. Bruni INAF - Istituto di
Astrofisica e Planetologia Spaziali, via Fosso del Cavaliere 100, 00133 Roma,
Italy  C. Sánchez-Fernández  European Space Astronomy Centre (ESA/ESAC),
Science Operations Department, 28691 Villanueva dela Cañada, Madrid, Spain  M.
Fiocchi  INAF - Istituto di Astrofisica e Planetologia Spaziali, via Fosso del
Cavaliere 100, 00133 Roma, Italy  L. Natalucci  INAF - Istituto di Astrofisica
e Planetologia Spaziali, via Fosso del Cavaliere 100, 00133 Roma, Italy  P.
Ubertini  INAF - Istituto di Astrofisica e Planetologia Spaziali, via Fosso
del Cavaliere 100, 00133 Roma, Italy

(Received -; Revised -; Accepted -)

###### Abstract

The microquasar MAXI J$1820+070$ went into outburst from mid-March until mid-
July 2018 with several faint rebrightenings afterwards. With a peak flux of
approximately 4 Crab in the $20-50$ keV, energy range the source was monitored
across the electromagnetic spectrum with detections from radio to hard X-ray
frequencies. Using these multi-wavelength observations, we analyzed quasi-
simultaneous observations from 12 April, near the peak of the outburst ($\sim
23$ March). Spectral analysis of the hard X-rays found a $kT_{e}\sim 30$ keV
and $\tau\sim 2$ with a CompTT model, indicative of an accreting black hole
binary in the hard state. The flat/inverted radio spectrum and the accretion
disk winds seen at optical wavelengths are also consistent with the hard
state. Then we constructed a spectral energy distribution spanning $\sim 12$
orders of magnitude using modelling in JetSeT. The model is composed of an
irradiated disk with a Compton hump and a leptonic jet with an acceleration
region and a synchrotron-dominated cooling region. JetSeT finds the spectrum
is dominated by jet emission up to approximately $10^{14}$ Hz after which disk
and coronal emission dominate. The acceleration region has a magnetic field of
$B\sim 1.6\times 10^{4}$ G, a cross section of $R\sim 2.8\times 10^{9}$ cm,
and a flat radio spectral shape naturally obtained from the synchroton cooling
of the accelerated electrons. The jet luminosity of $>8\times 10^{37}$ erg/s
($>0.15L_{Edd}$) compared to an accretion luminosity of $\sim 6\times 10^{37}$
erg/s, assuming a distance of 3 kpc. Because these two values are comparable,
it is possible the jet is powered predominately via accretion with only a
small contribution needed from the Blanford-Znajek mechanism from the
reportedly slowly spinning black hole.

Black Holes: individual (MAXI J$1820+070$) — X-rays: binaries — radiation
mechanisms: non-thermal

††journal: ApJ

##  1 Introduction

The term ”microquasar” was first applied to the persistent black hole
candidate (BHC) 1E $1740.7-2942$ after detecting radio jets from the known
hard X-ray source (Mirabel et al., 1992) that were similar to radio-loud
active galactic nuclei (AGNs). Jets were later found to be common features in
accreting BH systems in the hard state. Multi-wavelength studies showed
correlations between radio and X-ray luminosities (Gallo et al., 2003),
indicating a relationship between the emission mechanisms despite a large
physical separation between the two. Additionally, this correlation holds also
for supermassive BHs in AGN when accounting for mass (Merloni et al., 2003),
thus linking the mechanisms in stellar mass and supermassive BHs. Therefore
understanding the jet and X-ray components in microquasars can shed light on
AGN.

The low-mass X-ray binary MAXI J$1820+070$ (=ASASSN-18ey) was first detected
on 6.59 March 2018111http://www.astronomy.ohio-
state.edu/asassn/transients.html with the All-Sky Automated Survey for
SuperNovae (Shappee et al., 2014) and was detected $\sim 6$ days later by the
MAXI/GSC at 11 March 2018 19:48 UTC (Kawamuro et al., 2018). With a peak flux
of $\sim 4$ Crab in the $20-50$ keV energy band (Roques & Jourdain, 2019) and
a long decay, the source was a good candidate for numerous observing campaigns
across the electromagnetic (EM) spectrum (e.g. Muñoz-Darias et al. 2019;
Tucker et al. 2018; Stiele & Kong 2020; Bright et al. 2020) to explore various
aspects of the source. Combining observations from various campaigns enables
studying the various emission processes together.

Therefore we compiled quasi-simultaneous observations from public archives,
Astronomer’s Telegrams, and Gamma-ray Coordination Network Circulars to
construct the widest possible frequency range, we were able to find detections
covering nearly 12 orders of magnitude from the meter-wavelength frequencies
to hard X-rays on 12 April 2018 (MJD 58220). With this spectral energy
distribution (SED), we studied the spectral components independently before
investigating them jointly by constructing a model consisting of a leptonic
jet, an irradiated disk, and a corona, using the JetSeT
software222https://jetset.readthedocs.io/en/latest/.

Figure 1: (Top) The light curves for Swift/BAT $15-50$ keV (black diamonds),
$0.3-10$ keV Swift/XRT (green squares), and 4.7 GHz RATAN (blue triangles) for
the initial phase of the outburst. The time span of observations analyzed in
this work are denoted in red.

##  2 Observations

Figure 1 shows the initial period of MAXI J$1820+070$ outburst in several
wavelengths across the spectrum using data Swift/BAT (black diamonds),
Swift/XRT (green squares) (Stiele & Kong, 2020), and 4.7 GHz RATAN (blue
triangles) (Trushkin et al., 2018). The XRT and RATAN data have been
normalized to be on the same scale of the BAT data. The period of the
observations used in this work are bracketed in red.

In the following, we give information about the different simultaneous
observations collected from archives, covering different bands from radio to
gamma-ray on April 12, 2018. Further details can be found in Table 1.

###  2.1 JVLA

We retrieved calibrated Karl G. Jansky Very Large Array (VLA) data for
experiment VLA/18A-470 from the National Radio Astronomy (NRAO) online
archive. The JVLA antennas, in A configuration, were split in 3 subarrays in
order to obtain simultaneous observations at 6 different central frequencies
(4.7 GHz, 7.5 GHz, 8.5 GHz, 11 GHz, 20.7 GHz, 25.5 GHz). Data were imaged
using CASA (Common Astronomy Software Applications package) version
5.6.2333https://casa.nrao.edu following standard procedures.

###  2.2 ALMA

Atacama Large Millimiter/submillimiter Array (ALMA) data for project
2017.1.01103.T were retrieved from the ESO archive, and pipelined at the the
Italian node of the European ALMA Regional Centre (INAF-Istituto di
Radioastronomia, Bologna). Imaging was performed with CASA version 5.1.1,
separately for each one of the 4 spectral windows (spw) present in the data
(Band 7, spw 5, 7, 9, 11) corresponding to the following central frequencies:
336 GHz, 338 GHz, 348 GHz, 350 GHz (Bonato et al., 2018).

###  2.3 VLT/X-shooter

A number of observations of MAXI J$1820+070$ were performed with the X-shooter
spectrograph Vernet et al. (2011) in the framework of the ESO program
0101.D-0356(A). We retrieved the processed spectra obtained during the 2018
outburst on 12 April from the European Southern Observatory (ESO) archive
science portal. These data have been reduced by using the ESO X-shooter
pipeline V2.7.0 and cover the 3000-25000 Åwavelength range. The observations
were conduced in nodding configuration with the slit oriented at the
parallactic angle and using slit widths of 1.3$\arcsec\times$11,
1.2$\arcsec\times$11 and 1.2$\arcsec\times$11 for the UVB, VIS and NIR arm,
respectively. This configuration yields a spectral resolution
R=$\lambda$/$\Delta\lambda$ of 4100, 6500 and 4300 for the UVB, VIS and NIR
arm, respectively. The observing conditions where good with a seeing of
0.47$\arcsec$ and an average airmass of the source during the acquisition of
1.3$\arcsec$. The total exposure times are 1640 s, 1300 s and 1520 s for the
UVB, VIS, and NIR arm, respectively. The reduced spectra have been corrected
for the foreground extinction using the Cardelli function Cardelli et al.
(1989) with R(V)=3.1 and AV=0.627 mag (Schlafly & Finkbeiner, 2011, via the
NASA/IPAC Extragalactic Database (NED)).

In order to estimate the slit loss effect in the X-shooter spectra, we first
applied standard aperture photometry on the $i^{\prime}$ acquisition image
using the iraf task phot. The zero point was calibrated using the stars in the
Panoramic Survey Telescope and Rapid Response System (Pan-STARRS1 Flewelling
et al., 2016) catalog.

From the aperture photometry we obtain an $i^{\prime}$ band apparent magnitude
of mAB= (12.20 $\pm$ 0.11) mag, corrected for foreground extinction. The
derived flux at the filter central wavelength is $\lambda$Fi =
(2.01$\pm$0.2)$\times$10-10 erg s-1 cm-2 which is in agreement with the
average flux measured from the spectrum in the 7300-7600 Å wavelength range:
$\lambda$Fi=(1.8$\pm$0.7)$\times$10-10 erg s-1 cm-2.

###  2.4 XMM-Newton/EPIC-pn

XMM-Newton ToO observations were carried out from 2018-04-12 07:27:58 to
09:39:28 UTC (obsid 0820880501) using burst mode. The European Photon Imaging
Camera (EPIC)-pn data were analyzed using the standard procedures with the
Science Analysis System (SAS) software version
xmmsas_20190531_1155-18.0.0444https://www.cosmos.esa.int/web/xmm-newton/sas-
threads.

###  2.5 INTEGRAL

The INTErnational Gamma-Ray Astrophysics Laboratory (INTEGRAL) observed MAXI
J$1820+070$ every 2–3 days between March 16, and May 8, via a series of Target
of Opportunity (ToO) observations. For this work, we selected the data
covering the interval (UTC) 11 April 2018 23:41:01 to 12 April 2018 11
14:00:21 (INTEGRAL revolution 1941). Here we focus on the analysis of data
provided by the Integral Soft Gamma-Ray Imager (ISGRI; $18-1000$ keV) placed
on the upper layer of the detector plane of the Imager on Board the INTEGRAL
Satellite (IBIS) telescope (Ubertini et al. 2003) and by the Optical
Monitoring Camera (OMC) ($500-600$ nm) instruments. The data were analyzed
using the Offline Science Analysis software (OSA) v11.0 available at the
INTEGRAL Science Data Center
(ISDC).555https://www.isdc.unige.ch/integral/analysis We followed standard
analysis procedures.

Table 1: MAXI J$1820+070$ observations log  Instrument | Start Time (UTC) | Stop Time (UTC)
---|---|---
VLITE | 07:25:00 12-04-2018 | 13:07:00 12-04-2018.
JVLA | 07:15:00 12-04-2018 | 13:14:50 12-04-2018
ALMA | 08:13:18 12-04-2018 | 09:20:16 12-04-2018
X-shooter | 07:41:08 12-04-2018 | 08:15:48 12-04-2018
EPIC-pn | 07:27:58 12-04-2018 | 09:39:28 12-04-2018
OMC | 23:41:01 11-04-2018 | 14:00:21 12-04-2018
ISGRI | 23:41:01 11-04-2018 | 14:00:21 12-04-2018
Table 2: Collected flux densities for the jet modelling.  Instrument | Frequency | Flux density
---|---|---
| (Hz) | (Jy)
VLITE | 3.39E+08 | 0.033$\pm$5.3
JVLA | 4.70E+09 | 0.0469$\pm$0.0047
| 7.50E+09 | 0.0488$\pm$0.0029
| 8.50E+09 | 0.0479$\pm$0.0048
| 1.10E+10 | 0.0483$\pm$0.0048
| 2.07E+10 | 0.0525$\pm$0.0054
| 2.55E+10 | 0.0530$\pm$0.0054
ALMA | 3.36E+11 | 0.116$\pm$0.006
| 3.38E+11 | 0.114$\pm$0.006
| 3.48E+11 | 0.115$\pm$0.006
| 3.50E+11 | 0.110$\pm$0.006
X-shooter | 1.37E+14 | 0.0920$\pm$0.0002
| 1.81E+14 | 0.0757$\pm$0.0012
| 2.40E+14 | 0.0652$\pm$0.0003
| 2.86E+14 | 0.0634$\pm$0.0006
| 3.79E+14 | 0.0689$\pm$0.0007
| 4.00E+14 | 0.0709$\pm$0.0002
| 4.42E+14 | 0.0743$\pm$0.0003
| 4.87E+14 | 0.0775$\pm$0.0005
| 5.15E+14 | 0.0794$\pm$0.0002
| 5.32E+14 | 0.0803$\pm$0.0003
| 5.76E+14 | 0.0831$\pm$0.0005
| 6.27E+14 | 0.0868$\pm$0.0003
| 7.00E+14 | 0.0952$\pm$0.0004
OMC | 5.66E+14 | 0.0813$\pm$0.0071
Figure 2: Jet emission between radio and optical bands, as reconstructed from
VLITE, JVLA, ALMA, and X-shooter observations. The blue dashed line is a
broken power law, used to identify the synchrotron peak frequency and flux
density. The red line is a power-law fit of the most expanded region of the
jet, considered as a physically separated component. The optical flux from OMC
is also shown, although not considered for the fit.

##  3 Results and discussion

###  3.1 The compact jet emission

With the collected flux densities between radio and UV bands, we built the jet
SED. In addition to the JVLA and ALMA data mentioned above, we considered Very
Large Array Low-band Ionosphere and Transient Experiment (VLITE, Clarke et al.
2016), also collected on 12 April 2018 (Polisensky et al., 2018). We show in
Figure 2 data from JVLA, ALMA, X-shooter, and OMC. For X-shooter, we
considered only the part of the spectrum not affected by absorption/emission
features, and averaged values for each of these intervals to obtain continuum
flux density values. In this way, we calculated 12 photometric measurements
(Table 2) covering the interval from NIR to UV. The overall shape of the
X-shooter spectrum shows a break, resulting in a change of the spectral index,
at about 2$\times$1014 Hz. This is most probably the frequency at which the
broadband SED is no longer dominated by the jet synchrotron emission, while
the accretion disk thermal emission increases (see Sec. 4). The single value
from OMC is in good agreement with the X-shooter photometry.

We fitted the radio to optical data set with a broken-power law to identify
the synchrotron peak frequency and the spectral slopes of the optically thin
and thick regions (Russell et al., 2013). We used the Astropy BrokenPowerLaw1D
function (see Astropy Collaboration et al. 2013; Price-Whelan et al. 2018)
adopting the LevMarLSQFitter routine to perform a Levenberg-Marquardt least
squares statistic. We excluded from the fit the X-shooter points above the
third one (Optical and UV ranges) since they show a turn-up of the flux
density liekly due to accretion disk emission. Similarly, we did not consider
data points below 20 GHz, since they show a different slope, probably
belonging to a physically distinct (and more expanded) radio jet component. We
obtained a synchrotron peak frequency of 1.6$\pm$0.2$\times 10^{13}$ Hz. The
estimated slope for the optically thick part of the spectrum is
$\alpha_{thick}=0.28\pm 0.02$ and $\alpha_{thin}=-0.61\pm 0.01$ for the
optically thin one (adopting the convention $S\propto\nu^{\alpha}$, where $S$
is the flux density, $\nu$ the frequency, and $\alpha$ the spectral index).
The slope of the lower frequency radio SED (0.3$-$10 GHz), fitted with the
PowerLaw1D Astropy function, is $\alpha=0.11\pm 0.02$. In Sec. 4, a detailed
physical model of this source is presented and provides a more precise
estimate of the jet parameters.

Figure 3: Accretion disk wind absorption features in the VLT/X-shooter optical
spectrum. Normalised flux errors are shown as a cyan shaded area. Troughs
deeper than 3$\times$RMS are highlighted in purple. The RMS value is 0.0027
for the He I $\lambda$5876 region, while 0.0023 for the H$\alpha$ and He I
$\lambda$6678 regions. The grey dashed lines indicate the 1$\pm$RMS intervals.

###  3.2 Accretion disk winds

Muñoz-Darias et al. (2019) discovered accretion disk winds in MAXI J$1820+070$
during both the 2018 hard state rise and decay. Absorption wind signatures
were detected in the blue wings of He I $\lambda$5876 and $\lambda$6678
emission lines, reaching a maximum terminal velocity ($v_{t}$) of 1200 km/s in
one of the epochs. For H$\alpha$, both a blue-wing broadened emission line
profile, implying a wind component of 1800 km/s, and a superimposed absorption
trough with a $v_{t}$=1200 km/s were found. Those authors collected several
epochs from 15 March to 4 November 2018, allowing them to follow the evolution
of the winds from the hard state to the disappearance during soft state, and
back.

The VLT/X-shooter data presented in this work add an epoch to monitoring in
Muñoz-Darias et al. (2019), falling in an uncovered time window of one month
between 26 March and 23 April. We normalized these spectra by dividing them
for the continuum emission, fitted with a third order spline3 function by
using the IRAF task continuum. These spectra are rich with emission lines from
the UVB to the NIR arms. Following the Muñoz-Darias et al. (2019) analysis, we
explored the presence of wind signatures linked to the mentioned emission
lines (He I and H$\alpha$), and found a significant absorption on the left
wing of He I $\lambda$5876\. In Fig. 3 (left panel) we show the relative
portion of the spectrum, with the absorption features highlighted in purple.
We consider as bona-fide absorption troughs the ones with a dip of at least
three times the continuum RMS. A prominent He I absorption feature is visible
between -700 and -900 km/s, showing the same profile as the correspondent
emission line. This one has a $v_{t}$ of 880 km/s. Further blue-ward
absorption features are visible, but since they are narrower and not connected
to the previous ones, we consider those as not related to the accretion disk
wind. The same is true for the narrow absorption features detected blue-wards
of the H$\alpha$ emission line (Fig. 3, central panel). For the He I
$\lambda$6678 line, a single absorption trough is detected between -800 and
-850 km/s (Fig. 3, right panel) with a $v_{t}$ of -825 km/s.
During this period, we observe strong asymmetries in the emission lines which
are commonly observed in lines emitted from the disc, particularly in the He I
$\lambda$ 6678 and in the H$\alpha$. Therefore, we explored line profile
properties by applying multi-component Gaussian fits using the python packages
curvefit and leastsq. In Figure 4 we show the result of this analysis for
H$\alpha$ (left panel) and He I $\lambda$6678 (right panel). The H$\alpha$
line analysis has been performed in the wavelength region 644$-$670 nm, which
includes the feature of interest and the local continuum. In this case we have
ignored from the fit the He I emission line, which falls at the end of the
analyzed wavelength range. The H$\alpha$ profile is well modelled by two
narrow Gaussian components, and only one broad Gaussian component is needed to
fit the red wing of the emission line. We note the absence of a blue-shifted
broad wing, which has been observed in Muñoz-Darias et al. (2019), as well as
p-cygni profiles signatures. However a forest of narrow absorption lines is
clearly visible in the blue region of the H$\alpha$. The two narrow components
are characterized by a central wavelength ($\lambda_{c}$) and a full width at
half maximum (FWHM) of $\lambda_{c}$= (6574.4$\pm$0.4) Å FWHM = (389$\pm$20)
km/s and $\lambda_{c}$= (6559.8$\pm$ 0.3) Å FWHM = (914$\pm$18) km/s,
respectively. While the broad red wing is centered at $\lambda_{c}$=
(6583.0$\pm$3.0) Å and has FWHM=(2982$\pm$210) km/s. From the redshift of the
broad wing with respect to the H$\alpha$ rest frame wavelength we derive an
outflow velocity of $v$= (923$\pm$14) km/s, while the separation between the
two narrow components is $\sim$ 667 km/s.

For the He I $\lambda$6678 line analysis we used the wavelength region
$664-672$ nm, which includes also the local continuum but excludes the
wavelength range in which the H$\alpha$ falls. The He I $\lambda$6678 line
profile is well modelled by three Gaussian components. The first one is well
centered on the rest-frame He I wavelength with a $\lambda_{c}$= (6679.2 $\pm$
0.4) Å and has a FWHM=(536$\pm$54) km/s. The two remaining Gaussians are
blueshifted and redshifted of $\sim$450 km/s with respect to the first
component, and are characterized by $\lambda_{c}$= (6669.4 $\pm$ 0.2) Å, FWHM
= (307$\pm$12) km/s and $\lambda_{c}$= (6689.7 $\pm$ 0.3) Å, FWHM =
(295$\pm$29) km/s, respectively.

As a whole, the detected optical disk wind features show properties with in
between what was found in the hard and soft state epochs collected by Muñoz-
Darias et al. (2019), confirming the decreasing trend of the optical wind
between the two states of the source.

Figure 4: Fits of the emission line profiles for H$\alpha$ (left panel) and He
I $\lambda$6678 (right panel). The residuals of the fits are shown at the
bottom of each panel. The line profiles are modelled with Gaussian components
(colored dashed lines). The total fitting model is represented by the magenta
solid line.

###  3.3 Soft X-ray

We fitted the XMM Newton/EPIC-pn spectrum in the $0.5-12$ keV energy range in
XSPEC using a(Tbabs*Powerlaw model. With this model we derive the following
parameters: nH = $0.13\pm 0.04\times 10^{22}\textrm{ cm}^{-2},kT_{bb}=0.24\pm
0.03\textrm{ keV, and }\Gamma=1.65\pm 0.08$ with $\chi 2/\nu=1.00$.

###  3.4 Hard X-ray

We fitted the INTEGRAL/IBIS/ISGRI spectrum in the $30-400$ keV energy range. A
systematic 1.5 % error was added to the data, following OSA 11 standard
recommendations666https://www.isdc.unige.ch/integral/analysis. A power-law fit
to the data in XSPEC found a photon index of $\Gamma=2.41\pm 0.01$ and a
$\chi^{2}/\nu=71.80$. The spectrum deviates from a simple power-law model,
especially at high energies, with residuals suggesting a Comptonized spectrum.
Fitting the data with a CompTT model using a photon temperature of 0.24 keV
fixed to the $kT_{bb}$ value from XMM finds a better fit, with $k_{T}=36.4\pm
0.9\textrm{ keV, }\tau=1.27\pm 0.05,\textrm{ and }\chi^{2}/\nu=6.21$. When
including a reflecting component (Reflect) with the reflection fraction fixed
to 1, the fit improves to $\chi^{2}/\nu=3.45\textrm{ with }k_{T}=38\pm
1\textrm{ keV and }\tau=1.44\pm 0.06$. Following Roques & Jourdain (2019), a
cutoff power-law was added Reflect*(CompTT)+cutoff with $\Gamma=1.6$ and a
cutoff energy of 200 keV that improved the fit to $0.71$ and has fit
parameters $k_{T}=27\pm 4\textrm{ keV, }\tau=2.2$.

To characterize the X-ray spectrum, a joint fit was performed between the two
instruments spanning $0.5-400$ keV using the model
Tbabs*Reflect*(diskbb+CompTT)+Tbabs*cutoff found best-fit parameters of
$kt_{BB}=0.27\pm 0.01\textrm{ keV, }kT=27\pm 1\textrm{ keV, and }\tau=2.2\pm
0.1\textrm{ with }\chi^{2}/\nu=0.95$.

Using this joint spectrum, we calculated the accretion luminosity in the
$1-200$ kev energy range and found a value of $\sim 6\times 10^{37}$ erg/s for
a distance 3 kpc.

Table 3: Irradiated disk fit parameters.  | diskir | diskir+po
---|---|---
$kT_{disk}$(keV) | $0.116\pm 0.007$ | $0.122\pm 0.007$
$\Gamma$ | $1.78\pm 0.02$ | $1.70\pm 0.04$
$kT_{e}$ (keV) | $58\pm 4$ | $37\pm 4$
$L_{C}/L_{D}$ | $4.7\pm 0.5$ | $4.7\pm 0.6$
$f_{out}$ | $(1\pm 40)\times 10^{-7}$ | $(4\pm 15)\times 10^{-2}$
$\log(r_{out})$ | $3.45\pm 0.04$ | $3\pm 1$
$\Gamma_{po}$ | $-$ | $1.6\pm 0.3$
Normpo | $-$ | $1.0\pm 1.7$
$\chi^{2}/\nu$ | $1.30$ | $0.97$

####  3.4.1 IR $-$ Hard X-ray Spectrum

Subsequently, we fit our data from the near-IR to hard X-rays using an
irradiated disk model to compare with Shidatsu et al. (2018), which analyzed a
similar energy range using observations from 24 March. The irradiated disk
model accounts for the effects of the Comptonized emission on the accretion
disk and the soft-excess that is seen in the hard state (Gierliński et al.,
2009). Figure 5 shows the spectrum from $0.001-400$ keV with the diskir model
shown as a solid red line and the power-law component of the diskir+po model
as a dashed black line. The power law is used to model the high-energy cutoff
power-law component in the previous section. Table 3 contains the fit
parameters using a diskir model with and without an additional power-law
component. We found that including a power-law component improved the fit at
high energies and reduced the $\chi^{2}/\nu$ from 1.30 to $\chi^{2}/\nu=0.97$
with an f-test probability of $3.7\times 10^{-8}$.

The origin of the power-law component is unclear. As shown below in Figure 7,
the expected jet flux is too low for the component to be jet emission at those
energies. However, the emission could possibly be from Comptonization of non-
thermal electrons as in the case of GRS $1716-249$ (2020MNRAS.494..571B).

Figure 5: MAXI J$1820+070$ spectrum from $0.001-400$ keV. The diskir model is
shown with a solid red line and the po component is shown as a black dashed
line.

##  4 Broadband SED modelling

We modelled the broadband SED of MAXI J1820+070 using a combination of jet
leptonic models and irradiated disk and corona model implemented in Jets SED
modeler and fitting Tool (JetSeT) 777https://jetset.readthedocs.io/en/latest/
(Tramacere, 2020; Tramacere et al., 2011, 2009). A more accurate description
of the model is discussed in Tramacere (in prep.). We assume that the
optical/UV up to keV energies is dominated by disc irradiation and coronal
emission. The emission in the mm to optical region is dominated by the non-
thermal emission of leptons accelerated in the jet by shock and/or stochastic
acceleration, and we assume that the break at $\approx 1.5\times 10^{13}$ Hz
is due to the transition from the optically thin to the optically thick
synchrotron emission. The radio emission is dominated by the terminal part of
the jet that starts beyond the acceleration region and extends up to a
distance of $\approx 1\times 10^{15}$ cm according to Bright et al. (2020). A
schematic view of the model is provided in Fig. 6

###  4.1 Individual Model Components Description

In the following we describe the implementation of each model component.

####  4.1.1 Irradiated Disk and Hot Corona

To model the UV to hard-X-ray emission we have used the disk Comptonization
plus disk irradiation model, DiskIrrComp implemented in JetSeT. The
DiskIrrComp is based on the diskir model (Gierliński et al., 2009) and the
Comptonization model of Zdziarski et al. (2009). In detail, we assume that a
classical multi-temperature disk with an inner temperature $T_{Disk}$ and an
extension $R_{in}=3R_{S}$ to $R_{out}$, expressed by the dimensionless
parameters $r_{in}=R/R_{in}$ and $r_{out}=R/R_{out}$. The disk spectrum is
modified due to the reprocessing of irradiated emission from the disk itself
and from the corona Compotonization tail. The corona emission is described by
a power law with an exponential cutoff with a photon index $\Gamma_{Comp}$ and
a cut-off energy $E_{Comp}=kT_{e}$ where $kT_{e}$ is the corresponding
electron temperature. The Compton hump component is described by a power-law
with exponential cut-off with a photon index $\Gamma_{hump}$ and a cut-off
energy $E_{hump}$. We refer to this model as Comp. hump. The normalization of
the Compton tail component is parameterized as a fraction of the disk
luminosity $L_{Disk}$ according to $L_{Comp}^{ratio}=L_{C}/L_{Disk}$. The
total bolometric flux will be $L_{bol}=L_{Disk}+L_{rep}+L_{C}$, where
$L_{rep}$ represents the thermalized fraction $f_{in}$ of $L_{C}$ thermalized
within $r_{in}$ and $r_{irr}=R_{irr}/R_{in}$, where $R_{irr}$ is the radius of
the inner disk irradiated by the Compton tail. A fraction $f_{out}$ of the
bolometric luminosity will irradiate the outer disk. The irradiation creates a
shoulder with a spectral trend $f_{out}\propto L_{bol}\nu^{-1}$ that extends
between $\nu_{1}=3kT(r_{out})$ and $\nu_{2}=3kT(r_{t})$, where $r_{t}$ is the
transitional radius between gravitational and irradiation energy release. This
effect depends strongly on $r_{out}$ and $f_{out}$, and it is present even
without corona Comptonization, because it represents the disk self-
irradiation. The presence of a Comptonization component will provide a further
heating of the disk in the inner part modifying the pure gravitational
temperature profile.

####  4.1.2 Pre-acceleration and Acceleration Region

We assume that electrons in the pre-acceleration region close to the base of
the jet are described by a thermal plasma with cooling dominated by adiabatic
losses. Once the particles approach the acceleration region they are
accelerated under the effect of diffusive shock acceleration and/or stochastic
acceleration and the corresponding energy distribution can be modeled by a
power-law with a high-energy cutoff

$N_{e,acc}(\gamma)=N\gamma^{-s}\exp(-\gamma/\gamma_{cut})$ (1)

where the value of $\gamma_{cut}$ takes into account the balance between
cooling and acceleration terms. The index $s$ is dictated by the competition
of the acceleration time scales and escape time scales (Tramacere et al.,
2011). We assume that the acceleration region extends from $z_{acc}^{start}$
to $z_{acc}^{end}$, with cross section $R_{acc}$ equal to the average cross
section of the jet at $z=(z_{acc}^{start}+z_{acc}^{end})/2$, with
$z_{acc}^{end}-z_{acc}^{start}=2R_{acc}$. The emission from the acceleration
region is reproduced using the jet leptonic model Jet implemented in JetSeT,
and we refer to it as JetAcc (Tramacere, in prep.).

####  4.1.3 Radio Jet

To model the radio jet emission we have used the JetSeT multi-zone radio jet
model RadioJet. This model implements a continuous jet as a sum of $N_{c}$
single zones, following the approach of Kaiser (2006), where for each zone the
values of $R$ and $B$ are ruled by Eq. 3, and the particle density scales as

$N_{s,i}=N_{s,0}(z_{s,0}/z_{i})^{m_{N}}$ (2)

where $N_{s,0}$ is the initial density of emitters at the starting point of
the radio jet $z_{s,0}$, $z_{i}$ is the average position of the $i_{th}$
component, and $m_{N}$ is the index of the particle density law fixed to 2.
The initial particle density is a fraction $N_{frac}$ of that present in the
acceleration region and we fix it to 1. The radio jet extends from
$z_{radio}^{start}=(z_{acc}+R_{acc})K_{R}^{start}$ to
$z_{radio}^{end}(z_{acc}+R_{acc})K_{R}^{end}$, where $K_{R}^{start}$ and
$K_{R}^{end}$ are free parameters. In the present analysis we fix
$K_{R}^{start}=1$ , and $K_{R}^{end}$ is fixed in order to match the value of
$1\times 10^{15}$ cm according to the analysis presented in Bright et al.
(2020). The particle distribution in each region has the same spectral law as
in the acceleration region, but we decrease the value of $\gamma_{cut}$ to
take into account the effect of the cooling when the particles leave the
acceleration region. In our analysis we take into account only synchrotron
cooling and we evolve $\gamma_{cut}$ according to Eq. 27 in Kaiser (2006).
More details about the connection between the acceleration and radio are
discussed in Sec. 4.3

Figure 6: A schematic view of the jet model setup. The purple region
identifies the pre-acceleration region, the cyan region identifies the
acceleration region, and the green region identifies the radio jet. The $z$
axis on the bottom shows the starting and end end point of each region. The
acceleration region is assumed to be spherical with a radius equal to the jet
cross section. The vertical black lines in the radio jet region marks
qualitatively the division of region in slices.

###  4.2 Phenomenological Model Setup

As a first step we set the geometrical properties of the jet, i.e. we define
the extent of the pre-acceleration, acceleration, and radio emission sites,
and the values of the magnetic field. We assume that the jet is launched at
distance $z_{0}$ from the BH, with an initial cross section $R_{0}$, and that
the bulk Lorentz factor ($\Gamma_{jet}$) of the jet is constant over the full
jet extent. The acceleration region starts at a distance $z_{acc}^{start}$
with a width equal to jet cross section diameter $R_{acc}=2R(z_{acc})$, and we
treat it as a spherical region. The radio region starts at
$z^{start}_{radio}=K_{R}^{start}(z_{acc}+R_{acc}$) and ends at a distance
$z^{end}_{radio}=K_{R}^{end}(z_{acc}+R_{acc})$ (a scheme of the model is
presented in Fig. 6). According to Bright et al. (2020) we fix the distance of
jet from the observer to the value of $d=3$ kpc, the termination of the radio
jet to the value of $z_{end}\approx 1\times 10^{15}$ cm, and the value of the
beaming factor to
$\delta=[\Gamma_{jet}(1-\beta_{jet}\cos(\theta_{obs})]^{-1}\approx 2.2$, using
the values of $\beta_{jet}=0.89$ and $\theta_{obs}=63^{\circ}$ reported in
Bright et al. (2020). We assume a ballistic jet model (Kaiser, 2006; Begelman
et al., 1984) characterized by

$\displaystyle B(z)$ $\displaystyle\propto$ $\displaystyle
B_{0}(z_{0}/z)^{m_{B}}$ $\displaystyle R(z)$ $\displaystyle\propto$
$\displaystyle R_{0}(z/z_{0})^{m_{R}}$ (3) $\displaystyle N(z)$
$\displaystyle\propto$ $\displaystyle R_{0}(z_{0}/z)^{m_{N}}$

with $m_{B}\approx 1$ and $m_{R}=1$, and $m_{N}=2.0$. This choice assumes that
the jet is very close to the ballistic regime, with a magnetic field dominated
by the toroidal component and justifies the assumption that the bulk Lorentz
factor is constant along the jet.

We use a black hole mass of $M_{BH}=8M_{\sun}$. The jet luminosity $L_{jet}$
is linked to the Eddington luminosity ($L_{Edd}$) according to

$L_{jet}=\frac{1}{2}q_{jet}L_{Edd}$ (4)

where $L_{Edd}\approx 1.3\times 10^{38}(M_{BH}/M_{\sun})$ erg s-1 (Rybicki &
Lightman, 1986). It is worth noting that our $q_{jet}$ parameter is not linked
directly to the accretion efficiency process, because the jet powering could,
in principle, be supported also by other mechanisms such as the
Blandford–Znajek mechanism (Blandford & Znajek, 1977), that predicts
electromagnetic extraction of energy and angular momentum from magnetized
accretion disc surround a black hole. Hence, our $q_{jet}$ parameter should
not be used to infer or constrain the accretion efficiency, and will be
discussed in a more accurate physical context in Tramacere (in prep.).

We assume that the jet is launched at a distance $z=z_{0}$ from the BH with
$z_{0}=50R_{S}\approx 1.2\times 10^{8}$ cm, where $R_{S}=(2GM_{BH})/{c^{2}}$.
The launching jet position $z=z_{0}$, in the current analysis is assumed
constant, to reduce the model complexity, and it is chosen according to
reference values published in previous analysis (Vila & Romero, 2010). The
initial radius of the jet is set to $R(z_{0})=0.1z_{0}$, resulting in an
opening angle of $\theta_{open}\approx 5.7^{\circ}$. We impose that in the
launching region the entire jet power is in the form of magnetic energy

$L_{jet}=L_{B}(z_{0})=\pi
U_{B}(z_{0})R(z_{0})^{2}\Gamma_{jet}^{2}\beta_{jet}c$ (5)

where $U_{B}=B^{2}/(8\pi)$, and setting $q_{jet}=0.2$ we obtain $B_{0}\approx
6.8\times 10^{6}$ G.

The value of $m_{B}$ can be constrained from the spectral index of radio jet
emission, $\alpha_{R}\approx 0.15$, according to the Eq. 39 in Kaiser (2006)
that refers to the case of strong radiative cooling and almost constant value
of the electron distribution high-energy cutoff. According to this scenario,
that is very similar to what we expect in our case, we can rearrange Eq. 39 in
Kaiser (2006) as:

$m_{B}=\frac{1+m_{R}}{2-\alpha_{R}}$ (6)

that is similar to the trend of the thick radio spectrum discussed in Pe’er &
Casella (2009a), and we obtain a value of $m_{B}\approx 1.1$ We stress that
this is an initial guess done assuming that the jet is not changing after the
acceleration region. As we will discuss in the next section, during the model
fit we need to take into account that jet expansion might change above the
acceleration region, hence we will relax the constraint on $m_{B}$ and $m_{R}$
considered in the RadioJet emission.

To constrain the value of $z_{acc}$ we impose that $R_{acc}=R(z_{acc})$,
$B_{acc}=B(z_{acc})$ and $N_{e,acc}$ correspond to a synchrotron self-
absorption frequency of $\nu_{t}\approx 1.5\times 10^{13}$ Hz. This value of
$\nu_{t}$ is obtained from the phenomenological fit of the optically-thin to
optically-thick synchrotron emission between mm and optical data shown in Fig.
2. In order to solve this problem we combine the analytical expression of the
synchrotron self-absorption frequency ($\nu_{t}$) (Rybicki & Lightman, 1986),
evaluated at the peak i.e. $\alpha_{\nu}=0$

$\nu_{t}=\nu_{L}\Big{[}\frac{\pi\sqrt{\pi}}{4}\frac{qR_{acc}N_{e,acc}}{B_{acc}}f_{k}(s)\Big{]}^{\frac{2}{s+4}},$
(7)

and of that the synchrotron emissivity (Rybicki & Lightman, 1986)
$\epsilon_{s}(\nu)$ :

$\epsilon_{s}(\nu)=\frac{F_{\nu}d_{L}^{2}}{V}=\frac{3\sigma_{T}cN_{e,acc}U_{B}^{acc}}{16\pi\sqrt{(}\pi)\nu_{L}}f_{\epsilon}(s),$
(8)

where $q$ is the electron charge, $U_{B}^{acc}$ is the value of the magnetic
field, $V$ is the volume of a spherical geometry of volume $V$ of radius
$R_{acc}$, $s$ is the slope of the electron distribution power-law and
$\nu_{L}=\frac{qB}{2\pi m_{e}c}$ is the Larmor frequency, and where the
functions $f_{k}(s)$ and $f_{\epsilon}(s)$ are approximated to percent
accuracy as reported in Ghisellini (2013). The value of $s$ is obtained using
the optically thin spectral index $\approx 0.6$ from the phenomenological fit
in Fig. 2, according to the relation $s=2\alpha+1\approx 2.2$ (Rybicki &
Lightman, 1986). We solve Eq. 8 with respect to $N_{e}^{acc}$ and then
substitute in Eq. 7, and we insert the functional form of $B=B(z_{acc})$ and
$R=R(z_{acc})$ according to Eq. 3. The final equation solved with respect to
$z_{acc}$ reads:

$z_{acc}=\Big{[}\Big{(}\frac{\nu_{t}2\pi
m_{e}c}{qB_{0}^{2}z_{0}^{m_{B}}}\Big{)}^{\frac{s+4}{2}}\frac{3\sigma_{T}(B_{0}R_{0})^{2}f_{\epsilon}(s)z_{0}^{2\Delta_{m}}}{16r_{e}^{2}\pi^{3}f_{k}(s)F_{\nu}d_{L}^{2}}\Big{]}^{\psi}$
(9)

where $r_{e}=q^{2}/(m_{e}c^{2})$ is the classical electron radius,
$\Delta_{m}=m_{B}-m_{R}$, and $\psi=\frac{2}{4\Delta_{m}-m_{B}(s+4)}$.

Consequently, the starting position of the radio jet is set to
$z_{radio}^{start}=z_{acc}^{end}=z_{acc}+R_{acc}\approx 3.1\times 10^{10}$ cm,
with an extent derived from Bright et al. (2020) of $z_{end}\gtrsim
30000z_{radio}^{start}$

The value of the cut-off of the electron distribution is set to
$\gamma_{cut}=60$, in order to produce the peak of the synchrotron emission
above the IR frequencies for a magnetic field $B_{acc}\approx 1.8\times
10^{4}$ G, with a power-law slope $s\approx 2.1$ that is slightly lower then
the value derived from the optically thin spectral index.

The constrained value of $z_{acc}$ can be used to derive the hadroninc content
of the jet energetic in form of cold protons. Following Vila & Romero (2010)
we impose that in the acceleration region of the jet the magnetic energy of
the jet is in subequipartition with the bulk kinetic energy of the cold
protons, a condition that is mandatory to allow the mechanical compressibility
of the plasma (Komissarov et al., 2007). We define the parameter
$\rho^{acc}_{p,B}=U_{p}(z_{acc})/U_{B}(z_{acc})$, where
$U_{p}(z)=n_{p}(z)m_{p}c^{2}$, and we require that $\rho_{p,B}>1$. This choice
sets a value of cold proton luminosity in the acceleration region
$L_{p}(z_{acc})>3.6\times 10^{37}$ erg $s^{-1}$.

Table 4: Phenomenological Setup Parameters Input Parameters
---
par. name | units | input value
$z_{0}$ | cm | $1.12\times 10^{8}$
$r_{0}$ | cm | $1.12\times 10^{7}$
$M_{BH}$ | $M_{\sun}$ | 8
$q_{jet}$ |  | 0.20
$F_{\nu}^{t}$ | Jy | $0.5$
$\nu_{t}$ | Hz | $1.5\times 10^{13}$
$s$ |  | 2.1
$\rho^{acc}_{p,B}$ |  | $>1$
$m_{B}$ |  | 1.1
$m_{R}$ |  | 1.0
Output Parameters
par. name | units | output value
$B_{0}$ | G | $6.8\times 10^{6}$
$B_{acc}$ | G | $1.8\times 10^{-4}$
$L_{p}^{acc}$ | erg s-1 | $>3.6\times 10^{37}$
$z_{acc}^{start}$ | cm | $2.4\times 10^{10}$
$z_{acc}^{end}$ | cm | $2.9\times 10^{10}$
$z_{acc}$ | cm | $2.6\times 10^{10}$
$R_{acc}$ | cm | $2.6\times 10^{9}$
$z_{radio}^{start}$ | cm | $2.9\times 10^{10}$
$z_{radio}^{end}$ | cm | $\gtrsim 1\times 10^{15}$
Table 5: JetSeT best fit model parameters  model name | par. name | units | best fit value | error | starting value | fit boundaries | frozen
---|---|---|---|---|---|---|---
CompHump | $E_{hump}$ | keV | 26 | 14 | 20 | [ 15 ; 35] | False
” | $\Gamma_{hump}$ |  | -0.5 | 2 | -1.2 | [ -2 ; 2] | False
DiskIrrComp | $T_{Disk}$ | K |  |  | $1.55\times 10^{6}$ |  | True
” | $L_{Disk}$ | erg s-1 | $1.09\times 10^{37}$ | $1.0\times 10^{32}$ | $1\times 10^{37}$ | [ $1\times 10^{36}$ ; $1\times 10^{39}$] | False
” | $r_{out}$ |  | $3.58\times 10^{3}$ | $0.21\times 10^{3}$ | $5\times 10^{3}$ | [ 1 ; – ] | False
” | $r_{irr}$ |  |  |  | 1.1 |  | True
” | $\Gamma_{Comp}$ |  | 1.64 | 0.12 | 1.65 | [ 1.3 ; 1.9 ] | False
” | $E_{Comp}$ | keV | 150 | 100 | 140 | [ 20 ; 200 ] | False
” | $L_{Comp}^{ratio}$ |  | 4.1 | 0.6 | 4.5 | [ 0 ; – ] | False
” | $f_{in}$ |  |  |  | 0.1 |  | True
” | $f_{out}$ |  | $1\times 10^{-2}$ | $40\times 10^{-2}$ | 0.01 | [ 0 ; – ] | False
DiskIrrComp | $r_{out}$ |  | $3.4\times 10^{3}$ | $0.5\times 10^{3}$ | $3.58\times 10^{3}$ | [ 1 ; – ] | False
” | $f_{out}$ |  | $7.33\times 10^{-3}$ | $0.15\times 10^{-3}$ | $1\times 10^{-2}$ | [ 0 ; – ] | False
” | $L_{Comp}^{ratio}$ |  | 4.270 | 0.016 | 4.1 | [ 0 ; – ] | False
JetAcc | $N_{e,acc}$ | cm-3 | $9.998\times 10^{11}$ | $0.001\times 10^{11}$ | $1.0\times 10^{12}$ | [0 ; – ] | False
” | $s$ |  | 2.082 | 0.007 | 2.1 | [- ; - ] | False
” | $\gamma_{cut}$ |  | $65.4$ | $1.7$ | 60 | [1 ; – ] | False
” | $R_{acc}$ | cm | $2.6\times 10^{9}$ | $1.0\times 10^{1}$ | $2.6\times 10^{9}$ | [ $1.32\times 10^{9}$ ; $3.96\times 10^{9}$] | False
” | $z_{acc}$ | cm |  |  | $2.8\times 10^{10}$ |  | True
” | $B_{acc}$ | G | 17986 | $1.0\times 10^{-3}$ | 17986 | [ 8993 ; 26980 ] | False
” | $\theta_{jet}$ | deg |  |  | 63 |  | True
” | $\Gamma_{jet}$ |  |  |  | 2.19 |  | True
RadioJet | $z_{inj}$ |  |  |  | $2.5\times 10^{10}$ |  | True
” | $N_{frac}$ |  |  |  | 1 |  | True
” | $K_{R}^{start}$ |  |  |  | 1 |  | True
” | $K_{R}^{end}$ |  |  |  | 30000 |  | True
” | $m_{jet}$ |  | 1.203 | 0.001 | 1.1 | [ 0.5 ; 1.5 ] | False

###  4.3 Model Fit and Results

####  4.3.1 Initial model setup

To optimize the model we use the composite model interface FitModel provided
by JetSeT, that allows combining different models in a global model. This
model can be optimized by inserting it to the ModelMinimizer JetSeT plugin. In
the current analysis we use a frequentist approach and we use the Minuit
ModelMinimizer option. We have used the Data and ObsData JetSeT tools to
import the observed data, and we have added a 5% systematic error in the range
$[1\times 10^{8},1\times 10^{16}]$ Hz, to avoid that the large inhomogeneity
on the fractional error between radio and optical/UV data, could bias the fit
convergence. For the error estimate we provide only errors derived form the
MIGRAD module of Minuit, a more reliable estimate based on a Markov chain
Monte Carlo (MCMC) will be presented in Tramacere (in prep.)

The DiskIrrComp model, the Comp. hump model, and the JetAcc are independent,
on the contrary, JetAcc and radio RadioJet are bound.

Figure 7: The best-fit JetSeT model of the broadband SED. Top panel: the
$F_{\nu}$ representation of the global model fit. Bottom panel: the $\nu
F_{\nu}$ representation. The red line represents the global model, the dashed
lines correspond to the single components, the color is reported in the
legend. The best fit parameters are reported in Tab. 5. The residuals plot is
evaluated with respect to the $\nu F_{\nu}$ representation.

The initial values of the parameters for the DiskIrrComp model are chosen
according to the analysis presented in Sec 3.3 and Sec. 3.4. In detail, we set
the initial values of $L_{disk}=1\times 10^{37}$ erg s-1, of $r_{out}=5000$,
of $f_{out}=0.01$, and $L_{Comp}^{ratio}=4.5$ and we fix the inner disk
temperature to $T_{Disk}=1.55\times 10^{6}$K, and the parameters $r_{irr}=1.1$
and $f_{in}=0.1$, the choice adopted in Gierliński et al. (2009), when the
Comptonization of the outer disk is included in the irradiated disk.

For the JetAcc model, we fix $\theta_{jet}=63^{\circ}$, $\Gamma_{jet}=2.19$,
we put a relative bound of +/- 0.5 centered on the parameters values derived
in the previous section, $R_{acc}=2.6\times 10^{19}$ cm, and $B_{acc}\approx
1.8\times 10^{4}$ G, we freeze the initial value of $z_{acc}=2.6\times
10^{10}$ cm, and we leave free the parameters for the electron distribution.

The initial setup of the parameters of the RadioJet is more complex and we
need to take into account the physical connection with the acceleration region
and the cooling process. This effect plays a crucial role, indeed, as already
discussed in Kaiser (2006) and Pe’er & Casella (2009a), the combination of
synchrotron cooling and jet expansion (assuming a negligible contribution from
adiabatic cooling) will result in an asymptotic value of $\gamma_{cut}(t)$,
that can naturally explain the flat radio spectrum without the need to
introduce significant particle re-acceleration in the radio jet. We follow the
approach reported in Pe’er & Casella (2009a) (in the case of negligible
adiabatic cooling) and we set $m_{B}^{radio}=m_{R}^{radio}=m_{jet}$. The
particle cut-off evolution in the radio jet will evolve according to (Kaiser,
2006):

$\gamma_{cut}(t)=\frac{\gamma_{cut}}{1+\frac{\sigma_{T}B0^{2}}{6m_{e}c\pi(f)}\gamma_{cut}t_{0}^{1-f}(t^{f}-t_{inj}^{f})}$
(10)

where $f=1-2m_{jet}$, and $t_{0}=z_{0}/\beta_{jet}c\Gamma_{jet}$,
$t_{inj}=z_{inj}/\beta_{jet}c\Gamma_{jet}$ and
$t=z_{/}\beta_{jet}c\Gamma_{jet}$, are the comoving time scales. We freeze a
starting value of $z_{inj}=z_{acc}^{start}\approx 2.5\times 10^{10}$ cm.

Another effect to take into account is the fact that for $z>z_{acc}$ the
structure of the jet could change, for this reason we leave free the
parameters $m_{jet}$ with a fit boundary of [0.5,1.5], with an initial value
of 1.18, that is slightly larger than the value used for the phenomenological
constraining, but gives a better agreement with radio-to-optical data.

The density of emitters at the base of the RadioJet, is bound to be equal to
the density of emitters in the acceleration region $N_{e}$ calculated
according to Eq. 3, at $z=z_{radio}^{start}$, by fixing $N_{frac}=1.0$. We fix
the values of $K_{R}^{end}=3000$ and of $K_{R}^{start}=1$.

A list of the free and frozen and of the bounds is reported in Table 5 in the
columns ‘staring values’, ‘fit boundaries’, respectively.

####  4.3.2 Model fit results for the Disk and Corona emission

We fit first the DiskIrrComp and Comp. hump components restricting the fit
range to $\nu$= $[5\times 10^{14},10^{20}]$ Hz and we get $\chi^{2}=152$ for
98 degree of freedom ($N_{dof}$), corresponding to reduced
$\chi^{2}_{red}=1.55$. The parameters values are reported in the upper part of
Table 5.

The best-fit parameters resulting from JetSeT are similar to those obtained
form the XSPEC analysis for the diskir model. In particular the $r_{out}$
value ($3.58\times 10^{3}R_{in}$ and $3.45\times 10^{3}R_{in}$) and
$L_{C}/L_{D}$ (4.1 and 4.6), for JetSeT and XSPEC respectively. The $f_{out}$
parameter, is unconstrained both for XSPEC and the JetSeT. However, a well
constrained value is obtained when the jet component is added as shown in
section 4.3.3. Because the JetSeT model for the Comptonized emission is
phenomenological, the high-energy range of the irradiated disk is fit as a
cutoff power-law and thus is not directly comparable to the diskir parameters.
For that portion of the spectrum, JetSeT found $\Gamma=1.64$ (compared to 1.78
from diskir) and $E_{C}=150$ keV (compared to $kT_{e}=58$ keV from diskir). We
do note that $E_{C}/kT_{e}\approx 2.6$, which falls within the predicted range
of ratios between cutoff energy and electron temperature (Petrucci et al.,
2000, 2001), suggesting that the values are in agreement, even though the
incertitude on the JetSeT value is quite large.

####  4.3.3 Model fit results for the jet emission

To fit the full band SED we freeze all the parameters in the CompHump and
DiskIrrComp components, except for $r_{out}$, $f_{out}$, and $L_{C}/L_{D}$,
and we fit the global model over the full SED band in the range $\nu$=
$[5\times 10^{8},10^{20}]$ Hz .

The model fit converged with a final $\chi^{2}=181$ for 122 degree of freedom
($N_{dof}$), corresponding to a $\chi^{2}_{red}=1.48$. The parameters values
are reported in the bottom part of Table 5, and the parameters derived from
the best-fit model are reported in Table 6. Regarding the DiskIrrComp, we note
that adding the jet component results in a better constraint on the value of
$f_{out}$=$(7.33\pm 0.15)\times 10^{-3}$, that is in the expected range of
other black hole binaries in the hard state (Gierliński et al., 2009).
Moreover, restricting the fit statistics to the same interval used in the
XSPEC analysis, we get a $\chi^{2}=157$ with $N_{dof}=107$, corresponding to a
$\chi^{2}_{red}=1.6$. Regarding the jet component, we note that final best-fit
model parameters did not change significantly from the input values,
suggesting that the phenomenological setup was able to find a configuration
very close to the optimal one, even though the fit might be biased by the
degeneracy among some parameters. We will investigate this problem in a
forthcoming work Tramacere (in prep.), by means of Bayesian approach based on
the MCMC technique. In general, we find that our assumption based on the
connection between a compact acceleration region feeding the extended radio
jet is able to model self-consistently the UV-to-optical emission, reproducing
the observed flat radio spectrum. In particular, we find that, according to
our best fit model, the particles in the radio region reach the asymptotic
value of $\gamma_{cut}\approx 8$, and keep it almost constant as result of the
decrease in the cooling synchrotron cooling rate due to the jet expansion.
This behaviour is in agreement with the results of Pe’er & Casella (2009b) and
Kaiser (2006) for the case of synchrotron cooling dominating over the
adiabatic one. It is worth discussing some specific parameters in detail:

Table 6: Model parameters evaluated from the best-fit model par. name | units | value | setup value
---|---|---|---
$q_{jet}$ |  | $>0.15$ | 0.20
$U_{e}/U_{B}$ |  | 0.18 | –
$N^{acc}_{e}/N_{p}^{cold}$ |  | $<94$ | –
$L_{jet}^{acc}$ | erg s-1 | $>8.0\times 10^{37}$ | –
$L_{rad}^{acc}$ | erg s-1 | $1.1\times 10^{36}$ | –
$L_{B}^{acc}$ | erg s-1 | $3.6\times 10^{37}$ | $3.6\times 10^{37}$
$L_{e}^{acc}$ | erg s-1 | $6.6\times 10^{36}$ | –
$L_{p}^{acc}$ | erg s-1 | $>3.6\times 10^{37}$ | $>3.6\times 10^{37}$

  * •

$q_{jet}>0.15$. This value is compatible with the input value $q_{jet}=0.2$.
As already discussed in the previous section, the $q_{jet}$ parameter is not
linked directly to the accretion efficiency because the jet powering could, in
principle, be supported also by other mechanisms such as the Blandford–Znajek
mechanism Blandford & Znajek (1977), that takes into account advection of
magnetic flux from an accretion disk surrounding the Black Hole. Hence, our
$q_{jet}$ parameter can not be used to infer or constrain the accretion
efficiency.

  * •

$U_{e}/U_{B}=0.18$. The $U_{e}/U_{B}$ is not far from equipartition, and it is
obtained without providing any constraint. This proves that the combination of
the phenomenological model setup and the minimization of the global model
converged naturally toward a configuration close to the physical equipartition
of $U_{e}$ and $U_{B}$, giving further support to the choice of a compact
acceleration region that is connecting the pre-acceleration region to the
radio jet.

  * •

$N^{acc}_{e}/N_{p}^{cold}<94$. Since our model is leptonic, the content of
cold protons can be derived from ancillary conditions, as the condition that
the magnetic energy of the jet has to be in subequipartition with the bulk
kinetic energy of the cold protons, in order to allow the mechanical
compressibility of the plasma (Komissarov et al., 2007), and formation of
shocks/turbulent acceleration sites in the acceleration region. From the best
fit model we get that to respect the condition $\rho^{acc}_{p,B}>1.0$ we need
to impose a lower limit of the ratio of relativistic electrons to cold protons
$N_{e}/N_{p}^{cold}<112$. This value is compatible with the usual value of
$N_{e}/N_{p}^{cold}=10$ (Celotti & Ghisellini, 2008) used in the case of
relativistic jets with a leptonic radiative domination. Moreover, we note that
the value of $B_{acc}$ obtained from the best fit did not require a
significant change in the value of $L_{B}$ as derived from the
phenomenological model setup, and demonstrating that constraining $z_{acc}$
based on the value of $\nu_{t}$ is naturally in agreement with formation of
mechanical compression in the jet when $U_{p}>U_{B}$.

  * •

$m_{jet}=1.2$. The value of $m_{jet}$ is one the most critical, indeed it
dictates the topology and intensity of the magnetic field beyond the
acceleration region, and it is interesting to compare to the value of $m_{B}$
that is used to model the jet below the acceleration region. The initial guess
based on the value of $\alpha_{R}$ has required a small modification in order
to reproduce the observed radio spectrum, and the final model naturally
explains the almost flat radio spectrum as emission of the cooled electron
leaving the acceleration region.

##  5 Discussion and Conclusions

As MAXI J$1820+070$ was observed numerous times across the EM spectrum during
its outburst, there are multiple works relevant to portions of our multi-
wavelength analysis, though to date none study the source in such a complete
picture as is presented with our model from JetSeT.

Shidatsu et al. (2018) provides the most direct comparison to the analysis in
this work, though the source behavior is different before and after 26 March
(MJD 58206). They found that the optical and near-IR emission is not entirely
from disk emission and thus included a power-law to their diskir model with a
spectrum described by fixed parameters $kT_{disk}=0.35$ keV, $L_{C}/L_{D}=70$,
$f_{out}=5\times 10^{-3}$, and $R_{out}=10^{5}R_{in}$. Our fit found a
considerably lower $kT_{disk}$ (0.12 keV), $L_{C}/L_{D}$ (4.7), and $R_{out}$
($10^{3}R_{in}$), but a higher $f_{out}$ ($4\times 10^{-2}$). We note that the
source behavior between the two observations is different with changes in the
spectral hardness in hard X-rays (Roques & Jourdain, 2019), the development of
type-C QPOs (Stiele & Kong, 2020), and a reduction in the size of the corona
(Kara et al., 2019) that can possibly explain the differences.

Following the work of Muñoz-Darias et al. (2019), we explored the presence of
disk wind signatures in our VLT/X-shooter optical spectrum, as this data-set
falls between epochs 11 and 12 of Muñoz-Darias et al. (2019) campaign and adds
an epoch in their uncovered time window (between 26 March and 23 April). We
focus our spectral analysis on the He I $\lambda$ 5876, He I $\lambda$ 6678
and H$\alpha$ wavelength regions. We found shallow p-cygni profiles and strong
line asymmetries in all the three mentioned lines, while a broad outflow
component is detected only in the red wing of the H$\alpha$. Among the
observed absorption troughs, the one detected in the blue wing of He I
$\lambda$ 5876 is the more prominent and it results in a terminal wind
velocity $v_{t}$=880 km/s, which is consistent with the outflow velocity of
$v\sim$900 km/s, derived from the H$\alpha$ redshifted broad component. These
properties indicate that at this epoch optical disk winds are still present,
although with slower velocities with respect to what found in Muñoz-Darias et
al. (2019). The author also report an evolution of the line profiles during
their monitoring campaign and our observation confirms this trend. In
particular the observed H$\alpha$ profile can be interpreted as a continuation
in the evolving pattern of the line between the epochs 9 and 12 shown in
Figure 2 of Muñoz-Darias et al. (2019). Similar spectral variations were
previously reported by Tucker et al. (2018) and ascribed to the orbital motion
of the system. Interestingly, some of the most conspicuous optical wind
detections in Muñoz-Darias et al. (2019) occur in epochs corresponding to the
hard state of the source, when radio emission and strong jet activity are
present (Bright et al., 2020) and the peak of the optical outburst of the
source is reported. This led the authors to the conclusion that the optical
wind detected in MAXI J$1820+070$ is simultaneous with the jet. Our wind
signatures detection, together with the results from our broad band spectral
analysis are consistent with this scenario.

Our phenomenological analysis of the compact jet found the data could be
modelled by a broken power-law with $\alpha_{thick}=0.28\pm 0.02$ and
$\alpha_{thin}=-0.61\pm 0.01$. Combining observations from late March and
early April, Russell et al. (2018) performed a similar analysis and found
spectral indices of $\alpha_{thick}\sim 0.3$ and $\alpha_{thin}\sim-0.7$.
Building on Russell et al. (2018), Shidatsu et al. (2018) estimated a
transition frequency of $\sim 3\times 10^{13}$ Hz and a corresponding flux
density of $\sim 0.4$ Jy. From these values they determined $B\sim 1\times
10^{4}$ G and $R\sim 2\times 10^{9}$ cm using equations from Shidatsu et al.
(2011). Our model peaks at 1.6$\pm$0.2$\times 10^{13}$ Hz with a flux density
of $\sim 0.35$ Jy thus resulting in similar values.

These values are in agreement with the phenomenological setup and with the
best-fit model from JetSeT. In particular the JetSeT best-fit model gives a
magnetic field in the acceleration region of $\approx 1.8\times 10^{4}$ G, and
a region radius of $\approx 2.6\times 10^{9}$ cm.

The corresponding energy density of the magnetic field is $\approx 1.3\times
10^{7}\textrm{ erg/cm}^{3}$ compared to the values of $8\times
10^{6}\textrm{erg/cm}^{3}$ from Shidatsu et al. (2018).

Additionally, we identify a separate radio spectral components at frequencies
below $\sim$10 GHz, showing an inverted power-law spectrum with slope
$\alpha=0.11\pm 0.02$. Bright et al. (2020), collecting data from different
epochs of VLA, Multi-Element Radio Linked Interferometer Network (eMERLIN),
and Meer Karoo Array Telescope (MeerKAT) observations, could identify at least
one ejected component during the transition from the hard to the soft state
(mid-June to mid-September 2018). Though the source is unresolved down to a
sub-arcsec resolution in the VLA observations considered here (collected in a
previous epoch), the presence of an additional low-frequency spectral
component could suggest that the ejecta later detected by Bright et al. (2020)
were already present at a sub-pc scale during the April 12th 2018 epoch
considered here.

This component is represented in the JetSeT broadband model by the RadioJet
component, and stems naturally from the cooling of the accelerated particle
leaving the acceleration region. Interestingly we find that the best-fit index
$m_{jet}\approx 1.2$ predicts a radio spectral index of
$\alpha=1-1/m_{jet}\approx 0.166$ that is close to the value found in the
power-law fit. We note, that the small difference between the two values, is
due to the fact the JetSeT RadioJet model takes into account the data range
from radio-to-mm frequencies, differently from the power-law fit, whose range
extends up $\approx 10^{10}$ Hz.

In conclusion, our broadband analyses of MAXI J$1820+070$ found the source in
a hard state with parameters similar to what was reported by Shidatsu et al.
(2018) The JetSeT broadband model was able to reproduce the full SED taking
into account both the disk/corona emission, and the leptonic radiativelly
dominated relativistic jet contribution. We found that the relativistic jet
required a total energy of $L_{jet}\geq 8.0\times 10^{37}$ erg/s,
corresponding to 0.15 $L_{Edd}$. This value represents a lower limit, since we
assume that the hadronic content of the jets is only in terms of cold protons,
without a significant radiative contribution. The flat radio spectral shape
stems naturally from the synchroton cooling of the electrons in the
acceleration regions, in agreement with previous analyses (Kaiser, 2006; Pe’er
& Casella, 2009a). In comparison, the accretion luminosity ($6\times 10^{37}$
erg/s) is comparable to the lower limit of the jet luminosity. Thus in MAXI
J$1820+070$, it is possible for the jet to be powered predominately via
accretion with only a small contribution from the Blanford-Znajek mechanism,
which in this case cannot provide much power since the black hole spin is
reported to be low (Bassi et al., 2020; Zhao et al., 2020).

We thank the Italian node of the European ALMA Regional Centre (ARC) for the
support. JR and GB acknowledge financial support under the INTEGRAL ASI-INAF
agreement 2019-35-HH.0 and ASI/INAF n. 2017-14-H.0. FO acknowledge the support
of the H2020 European Hemera program, grant agreement No 730970. The research
leading to these results has received funding from the European Union’s
Horizon 2020 Programme under the AHEAD2020 project (grant agreement n. 871158)
F.O. acknowledges the support of the H2020 European Hemera program, grant
agreement No 730970, and the support of the GRAWITA/PRIN-MIUR project: ”The
new frontier of the Multi-Messenger Astrophysics: follow-up of electromagnetic
transient counterparts of gravitational wave sources”. Based on observations
with INTEGRAL, an ESA project with instruments and science data centre funded
by ESA member states (especially the PI countries: Denmark, France, Germany,
Italy, Switzerland, Spain) and with the participation of Russia and the USA.
This research has made use of the services of the ESO Science Archive
Facility. Based on observations collected at the European Southern Observatory
under ESO programmes 2017.1.01103.T (ALMA) and 0101.D-0356(A) (VLT). ALMA is a
partnership of ESO (representing its member states), NSF (USA), and NINS
(Japan), together with NRC (Canada) and NSC and ASIAA (Taiwan), in co-
operation 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.

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