#  Generalized Mathisson-Papapetrou-Tulczyjew-Dixon Equations

Takayoshi Ootsuka<EMAIL_ADDRESS>Department of Physics,
Meisei University, 2-1-1 Hodokubo Hino, Tokyo191-8506, Japan  Ryoko Yahagi
<EMAIL_ADDRESS>Department of Physics, Faculty of Science, Tokyo
University of Science, 1-3 Kagurazaka, Shinjuku-ku, Tokyo, Japan

###### Abstract

We derive two generalizations of Mathisson-Papapetrou-Tulczyjew-Dixon
equations from Casalbuoni-Brink-Schwarz type pseudoclassical Lagrangians of
Majorana spinors on a Riemann-Cartan spacetime. One has a “color” freedom,
which makes the equations of motion also be a generalization of Wong
equations. The other is a spinor model coupled with a Rarita-Schwinger field
to preserve the supersymmetry. The coupling to the torsion is modified due to
the existence of the Rarita-Schwinger field. In both extensions, the Tulczyjew
condition is automatically satisfied.

##  I Introduction

The equations of motion of an extended body on a curved spacetime were first
considered by Mathisson in 1937 Mathisson . Subsequently, other contributions
were made by Papapetrou Papapetrou , Tulczyjew Tulczyjew , and Dixon Dixon .
We call the equations Mathisson-Papapetrou-Tulczyjew-Dixon (MPTD) equations in
this paper. A good collection of works and applications concerning the MPTD
equations is compiled in Puetzfeld .

Lagrangian point of view for the equations has also been discussed in some
literature Leclerc ; Lompay ; Unal ; Ramirez . Especially, Leclerc Leclerc
and Ünal Unal  have considered a spinning particle on a Riemann-Cartan
spacetime and appended a term depending on torsion to the original MPTD
equations. Barducci et al. Barducci  have used a pseudoclassical approach to
address the MPTD equations. A pseudoclassical model has Grassmann coordinates
which are a vestige of a spinor quantum field in the semiclassical limit. One
of the simplest models of a pseudoclassical particle is given by Cassalbuoni
Casalbuoni  and Brink and Schwarz Brink-Schwarz . The Lagrangian of this model
comprises Majorana spinors which represents the spin degrees of freedom of the
particle. In this paper, we extend this model, which is originally defined on
the Minkowski spacetime, to the one on a Riemann-Cartan spacetime introducing
the spin connection. Unlike Barducci , our Lagrangian is in a form of simple
square root. Two-dimensinal model has been discussed in our previous study
OITY .

In the following section, we introduce general notations used in this paper
and the first model that derives the combination of MPTD and Wong equations.
In section 3, the second model with the Rarita-Schwinger field is discussed.
Note that we use left derivatives throughout this paper.

##  II MPTD-Wong equations

###  II.1 Model

Our pseudoclassical Lagrangian is defined on a supermanifold, where both
Grassmann odd and even coordinates are considered. The notation
$x^{\mu},\,\mu=0,1,2,3$ is used for the Grassmann even coordinates which
describe the spacetime. For Grassmann odd coordinates, we use the real values
$\xi^{A\alpha},\,A=0,1,2,3,\,\alpha=1,\cdots,N$, where the index $\alpha$
represents an additional $N$-state internal degree of freedom, which we call
“color” of the pseudoclassical particle. The $(4,4N)$-dimensional
supermanifold concerned is denoted as $M^{(4,4N)}$. The submanifold
$M^{(4,0)}$ is a Riemann-Cartan spacetime with a metric $g$ with signature
$(+,-,-,-)$ and a spin connection
$\displaystyle\omega=\frac{1}{4}\omega_{ab}\gamma^{ab}=\omega_{c}\theta^{c}=\frac{1}{4}\omega_{abc}\gamma^{ab}\theta^{c},\,a,b,c\cdots=0,1,2,3$.
We define
$\displaystyle\gamma^{ab}=\frac{1}{2}\left(\gamma^{a}\gamma^{b}-\gamma^{b}\gamma^{a}\right)$
where $\gamma^{a}$ are gamma matrices, and
$\theta^{a}=e^{a}{}_{\mu}(x)dx^{\mu}$ where $e^{a}{}_{\mu}(x)$ are vierbeins.

We consider a pseudoclassical Lagrangian given by

$\displaystyle
L=-m\sqrt{\eta_{ab}\Pi^{a}\Pi^{b}},\quad\Pi^{a}=\theta^{a}+\langle\xi|\gamma^{5}\gamma^{a}|D\xi\rangle,$
(1)

where

$\displaystyle|D\xi\rangle=$
$\displaystyle|d\xi\rangle+\omega|\xi\rangle+A|\xi\rangle+i\kappa\theta_{a}\gamma^{a}|\xi\rangle$
$\displaystyle=$ $\displaystyle
e_{\alpha}\otimes|_{A}\rangle\left(d\xi^{A\alpha}+\frac{1}{4}\omega_{abc}(\gamma^{ab})^{A}{}_{B}\theta^{c}\xi^{B\alpha}+A^{\alpha}{}_{\beta
a}\theta^{a}\xi^{A\beta}+i\kappa\theta_{a}(\gamma^{a})^{A}{}_{B}\xi^{B\alpha}\right).$
(2)

Here, $m$ is the mass, the speed of light is set to $1$ and
$\gamma^{5}=i\gamma^{0}\gamma^{1}\gamma^{2}\gamma^{3}$. Note that this
Lagrangian can be also considered as a Finsler metric on a supermanifold
$M^{(4,4N)}$, which plays an important role to define the Lie derivative in
the next subsection. By introducing the constant $\kappa$, we have the
renormalized mass $m^{\prime}:=m(1+i\kappa\langle\xi|\gamma^{5}|\xi\rangle)$.
Accordingly, we take the bare and renormalized momentum as

$\displaystyle
P_{a}:=\frac{\partial{L}}{\partial{\Pi^{a}}}=m^{2}\frac{\Pi_{a}}{L},\quad
P_{a}P^{a}=m^{2},$ (3)
$\displaystyle\tilde{P}_{a}:=P_{a}(1+i\kappa\langle\xi|\gamma^{5}|\xi\rangle)=\frac{m^{\prime}}{m}P_{a},\quad\tilde{P}_{a}\tilde{P}^{a}=(m^{\prime})^{2}.$
(4)

If $\xi^{A\alpha}=0$, it is the normal Lagrangian of a relativistic particle.
$\xi^{A\alpha}$ are coefficients of a spinor
$|\xi\rangle=e_{\alpha}\otimes|_{A}\rangle\xi^{A\alpha}$ with respect to the
spinor basis $|_{A}\rangle$ and the basis $e_{\alpha}$ of a real
representation space of the Lie algebra conserned. We assume the
representation space has an inner product $\langle
e_{\alpha},e_{\beta}\rangle=h_{\alpha\beta}$ (similarly, $\langle
e^{\alpha},e^{\beta}\rangle=h^{\alpha\beta}$ and $\langle
e_{\alpha},e^{\beta}\rangle=\delta_{\alpha}^{\beta}$ for the dual basis
$e^{\alpha}$). We also introduce a gauge field $A=A^{i}T_{i}$ which acts on a
spinor $|\xi\rangle$ as

$\displaystyle A|\xi\rangle=e_{\alpha}\otimes|_{A}\rangle
A^{\alpha}{}_{\beta}\xi^{A\beta}=e_{\alpha}\otimes|_{A}\rangle
A^{\alpha}{}_{\beta a}\theta^{a}\xi^{A\beta},\quad
A^{\alpha}{}_{\beta}:=A^{i}(T_{i})^{\alpha}{}_{\beta}=A^{i}{}_{a}\theta^{a}(T_{i})^{\alpha}{}_{\beta},$
(5)

where $T_{i}$ form the basis of the Lie algebra for the model. In this
representation, the basis $T_{i}$ satisfies
$(T_{i})_{\alpha\beta}=-(T_{i})_{\beta\alpha}$. By introducing an anti-
symmetric spinor metric $B_{AB}$ Chev ; Suz  which has the properties

$\displaystyle B\gamma^{a}B^{-1}=\,^{t}\gamma^{a},\quad B_{AB}=-B_{BA},\quad
B^{2}=1,$ (6)

the coefficients of the metric dual spinor are determined as
$\xi_{A\alpha}=\xi^{B\beta}B_{BA}h_{\alpha\beta}$. The dual spinor is defined
by $\langle\xi|=\xi_{A\alpha}e^{\alpha}\otimes\langle^{A}|$, where
$\langle^{A}|$ is the dual basis of $|_{A}\rangle$. The inner product between
spinors $|\xi\rangle$ and $|\xi^{\prime}\rangle$ becomes

$\displaystyle\langle\xi|\xi^{\prime}\rangle=\xi_{A\alpha}\xi^{\prime
A\alpha}=\xi^{A\alpha}B_{AB}h_{\alpha\beta}\xi^{\prime
B\beta}=-\xi^{A\alpha}\xi^{\prime}_{A\alpha}.$ (7)

Due to the definition of the matrix $B_{AB}$, gamma matrices
$(\gamma^{a})_{AB}:={(\gamma^{a})^{C}}_{B}B_{CA}$ admit the following
properties:

$\displaystyle(\gamma^{a})_{AB}=-(\gamma^{a})_{BA},\hskip
28.45274pt(\gamma^{5})_{AB}=-(\gamma^{5})_{BA},$
$\displaystyle(\gamma^{ab})_{AB}:=\frac{1}{2}\left(\gamma^{a}\gamma^{b}-\gamma^{b}\gamma^{a}\right)_{AB}=(\gamma^{ab})_{BA},\hskip
28.45274pt(\gamma^{5}\gamma^{a})_{AB}=(\gamma^{5}\gamma^{a})_{BA},$
$\displaystyle(\gamma^{abc})_{AB}:=\frac{1}{6}\left(\gamma^{a}\gamma^{b}\gamma^{c}+\gamma^{b}\gamma^{c}\gamma^{a}+\gamma^{c}\gamma^{a}\gamma^{b}-\gamma^{a}\gamma^{c}\gamma^{b}-\gamma^{c}\gamma^{b}\gamma^{a}-\gamma^{b}\gamma^{a}\gamma^{c}\right)_{AB}=(\gamma^{abc})_{BA},$
$\displaystyle(\gamma^{5}\gamma^{ab})_{AB}=(\gamma^{5}\gamma^{ab})_{BA},\hskip
14.22636pt(\gamma^{5}\gamma^{abc})_{AB}=-(\gamma^{5}\gamma^{abc})_{BA}.$ (8)

We also use the commutation relations

$\displaystyle\left[\frac{\gamma^{ab}}{2},\gamma^{c}\right]=$
$\displaystyle\left(\eta^{bc}\gamma^{a}-\eta^{ac}\gamma^{b}\right)$ (9)
$\displaystyle\left[\frac{\gamma^{ab}}{2},\frac{\gamma^{cd}}{2}\right]=$
$\displaystyle\left(\eta^{bc}\frac{\gamma^{ad}}{2}-\eta^{ac}\frac{\gamma^{bd}}{2}-\eta^{bd}\frac{\gamma^{ac}}{2}+\eta^{ad}\frac{\gamma^{bc}}{2}\right)$
(10)

in the rest of the discussion.

###  II.2 Vector fields and the equations for their conjugate quantities

Let $\displaystyle v=v^{I}\frac{\partial{}}{\partial{z^{I}}}$ be a vector
field on $M^{(4,4N)}$. Here, we set $(z^{I}):=(x^{\mu},\xi^{A\alpha})$, where
capital Roman letter $I$ stands for both spacetime and spinor indices. We
define Lie derivative of the Lagrangian (Finsler metric) with respect to $v$
Killing  as

$\displaystyle{\cal
L}_{v}L:=v^{I}\frac{\partial{L}}{\partial{z^{I}}}+dv^{I}\frac{\partial{L}}{\partial{dz^{I}}}=d\left(v^{I}\frac{\partial{L}}{\partial{dz^{I}}}\right)+v^{I}\left\\{\frac{\partial{L}}{\partial{z^{I}}}-d\left(\frac{\partial{L}}{\partial{dz^{I}}}\right)\right\\}.$
(11)

The derivative $d$ here is a total derivative that is calculated as

$\displaystyle
df(z,dz)=dz^{I}\frac{\partial{f}}{\partial{z^{I}}}+d^{2}z^{I}\frac{\partial{f}}{\partial{dz^{I}}},$
(12)

for a function $f(z,dz)$. When the Euler-Lagrange equations
$\displaystyle\frac{\partial{L}}{\partial{z^{I}}}-d\left(\frac{\partial{L}}{\partial{dz^{I}}}\right)=0$
are satisfied, this relation is equivalent to the identity

$\displaystyle
v^{I}\frac{\partial{L}}{\partial{z^{I}}}+dv^{I}\frac{\partial{L}}{\partial{dz^{I}}}=d\left(v^{I}\frac{\partial{L}}{\partial{dz^{I}}}\right),$
(13)

for any $v^{I}$. This is the evolution equation of the quantity $\displaystyle
v^{I}\frac{\partial{L}}{\partial{dz^{I}}}$. In the following discussion, the
relation (13) is repeatedly used with various vector fields. To specify the
physical meaning of the quantity $\displaystyle
v^{I}\frac{\partial{L}}{\partial{dz^{I}}}$, we also mention the related
symmetry beforehand.

####  II.2.1 Color symmetry

Firstly, we show the color symmetry. The gauge transformation is expressed as

$\displaystyle\delta|\xi\rangle=\epsilon|\xi\rangle\quad\left(\delta\xi^{A\alpha}=\epsilon^{\alpha}{}_{\beta}\xi^{A\beta}\right),\hskip
14.22636pt\delta A=-D\epsilon=-\left(d\epsilon+[A,\epsilon]\right),\hskip
14.22636pt\epsilon^{\alpha}{}_{\beta}(x):=\epsilon^{i}(x)(T_{i})^{\alpha}{}_{\beta}$
(14)

where $\epsilon^{i}(x)$ are arbitrary functions of $x^{\mu}$ and
$\epsilon_{\alpha\beta}=-\epsilon_{\beta\alpha}$. Accordingly, we have

$\displaystyle\delta\langle\xi|=-\langle\xi|\epsilon,\quad\delta|D\xi\rangle=\epsilon|D\xi\rangle,$
(15)

which leads to $\delta\Pi^{a}=0$. Thus, the Lagrangian is invariant under this
transformation.

The corresponding vector field is

$\displaystyle
v=\epsilon^{\alpha}{}_{\beta}\xi^{A\beta}\frac{\partial{}}{\partial{\xi^{A\alpha}}},$
(16)

where $\epsilon^{\alpha}{}_{\beta}$ are set to be antisymmetric constants from
now on. Then, the equation (13) becomes

$\displaystyle\epsilon^{\alpha}{}_{\beta}\xi^{A\beta}\frac{\partial{L}}{\partial{\xi^{A\alpha}}}+\epsilon^{\alpha}{}_{\beta}d\xi^{A\beta}\frac{\partial{L}}{\partial{d\xi^{A\alpha}}}=d\biggl{[}\epsilon^{\alpha}{}_{\beta}\xi^{A\beta}\frac{\partial{L}}{\partial{d\xi^{A\alpha}}}\biggr{]},$
(17)

i.e.,

$\displaystyle-\frac{1}{2}\epsilon_{\alpha\beta}\left(A^{\alpha}{}_{\gamma}J^{\gamma\beta}+A^{\beta}{}_{\gamma}J^{\alpha\gamma}\right)=\frac{1}{2}\epsilon_{\alpha\beta}dJ^{\alpha\beta},$
(18)

where we can define

$\displaystyle
J^{\alpha\beta}:=2P_{a}\langle\xi|\gamma^{5}\gamma^{a}|_{A}^{\alpha}\rangle\xi^{A\beta}=2P_{a}\xi_{A}^{\alpha}(\gamma^{5}\gamma^{a})^{A}{}_{B}\xi^{B\beta}$
(19)

and identify it as a color corrent. Eliminating $\epsilon_{\alpha\beta}$ from
both sides, we have the covariant conservation of the color current
$J^{\alpha\beta}$:

$\displaystyle
DJ^{\alpha\beta}:=dJ^{\alpha\beta}+A^{\alpha}{}_{\gamma}J^{\gamma\beta}+A^{\beta}{}_{\gamma}J^{\alpha\gamma}=0.$
(20)

This is one of Wong equations. With the equation (20), it is easy to show

$\displaystyle d\left(J_{\alpha\beta}J^{\alpha\beta}\right)=0,$ (21)

i.e., the norm of $J^{\alpha\beta}$ is conserved.

####  II.2.2 Lorentz transformation

Lorentz transformation is defined by

$\displaystyle\delta\theta^{a}=\epsilon^{a}{}_{b}\theta^{b},\hskip
14.22636pt\delta|\xi\rangle=\frac{1}{4}\epsilon_{ab}\gamma^{ab}|\xi\rangle,\hskip
14.22636pt\delta\omega_{ab}=-d\epsilon_{ab}-\omega_{ac}\epsilon^{c}{}_{b}+\epsilon_{ac}\omega^{c}{}_{b},$
(22)

where $\epsilon_{ab}(x)$ are antisymmetric functions of $x^{\mu}$. Using
identities (9) and (10), we obtain

$\displaystyle\delta\langle\xi|=-\frac{1}{4}\epsilon_{ab}\langle\xi|\gamma^{ab},\quad\delta|D\xi\rangle=\frac{1}{4}\epsilon_{ab}\gamma^{ab}|D\xi\rangle.$
(23)

Then, we have

$\displaystyle\delta\biggl{(}\eta_{ab}\Pi^{a}\Pi^{b}\biggr{)}=2\eta_{ab}\Pi^{a}\delta\Pi^{b}=2\eta_{ab}\epsilon^{b}{}_{c}\Pi^{a}\Pi^{c}=0,$
(24)

which shows the Lorentz invariance of the Lagrangian.

The vector field that corresponds to the Lorentz transformation is

$\displaystyle
v=\frac{1}{4}\epsilon_{ab}(\gamma^{ab})^{A}{}_{B}\xi^{B\alpha}\frac{\partial{}}{\partial{\xi^{A\alpha}}},$
(25)

where $\epsilon_{ab}$ are now set to be constant for the following argument.
The equation (13) for the above vector field becomes

$\displaystyle\frac{1}{4}\epsilon_{ab}(\gamma^{ab})^{A}{}_{B}\xi^{B\alpha}\frac{\partial{L}}{\partial{\xi^{A\alpha}}}+\frac{1}{4}\epsilon_{ab}(\gamma^{ab})^{A}{}_{B}d\xi^{B\alpha}\frac{\partial{L}}{\partial{d\xi^{A\alpha}}}=d\biggl{[}\frac{1}{4}\epsilon_{ab}(\gamma^{ab})^{A}{}_{B}\xi^{B\alpha}\frac{\partial{L}}{\partial{d\xi^{A\alpha}}}\biggr{]},$
(26)

i.e.,

$\displaystyle\epsilon_{ab}\biggl{(}P^{a}\langle\xi|\gamma^{5}\gamma^{b}|D\xi\rangle-\omega^{a}{}_{c}S^{bc}-i\kappa\theta^{b}P^{a}\langle\xi|\gamma^{5}|\xi\rangle\biggr{)}$
$\displaystyle=-\frac{1}{2}\epsilon_{ab}dS^{ab},$ (27)

where

$\displaystyle
S^{ab}:=-\frac{1}{2}P_{c}\langle\xi|\gamma^{5}\gamma^{abc}|\xi\rangle$ (28)

is the spin of the particle. Defining the covariant derivative of the spin

$\displaystyle
DS^{ab}:=dS^{ab}+\omega^{a}{}_{c}S^{cb}+\omega^{b}{}_{c}S^{ac},$ (29)

we obtain

$\displaystyle DS^{ab}=\tilde{P}^{a}\theta^{b}-\tilde{P}^{b}\theta^{a},$ (30)

where an identity

$\displaystyle\tilde{P}^{a}\theta^{b}-\tilde{P}^{b}\theta^{a}=-P^{a}\langle\xi|\gamma^{5}\gamma^{b}|D\xi\rangle+P^{b}\langle\xi|\gamma^{5}\gamma^{a}|D\xi\rangle+i\kappa\theta^{b}P^{a}\langle\xi|\gamma^{5}|\xi\rangle-i\kappa\theta^{a}P^{b}\langle\xi|\gamma^{5}|\xi\rangle$
(31)

is used. The equation (30) is the very one of the MPTD equations. The
Tulczyjew condition

$\displaystyle S^{ab}\tilde{P}_{b}=0$ (32)

is automatically satisfied and the norm of the spin is conserved:

$\displaystyle d\left(S_{ab}S^{ab}\right)=0.$ (33)

####  II.2.3 Translation

The vector field for the spacetime translation $\delta
x^{\mu}=\epsilon^{\mu}$, where $\epsilon^{\mu}$ are constants, is defined as

$\displaystyle v=\epsilon^{\mu}\frac{\partial{}}{\partial{x^{\mu}}}.$ (34)

The equation (13) is

$\displaystyle\epsilon^{\mu}\frac{\partial{L}}{\partial{x^{\mu}}}+d\epsilon^{\mu}\frac{\partial{L}}{\partial{dx^{\mu}}}=d\biggl{[}\epsilon^{\mu}\frac{\partial{L}}{\partial{dx^{\mu}}}\biggr{]},$
(35)

i.e.,

$\displaystyle\epsilon^{\mu}\biggl{(}\tilde{P}_{a}\partial_{\mu}e^{a}{}_{\nu}dx^{\nu}-\frac{1}{2}S^{bc}\partial_{\mu}\omega_{bc\nu}dx^{\nu}+\frac{1}{2}J^{\alpha\beta}\partial_{\mu}A_{\alpha\beta\nu}dx^{\nu}\biggr{)}=\epsilon^{\mu}d\biggl{(}\tilde{P}_{a}e^{a}{}_{\mu}-\frac{1}{2}S^{bc}\omega_{bc\mu}+\frac{1}{2}J^{\alpha\beta}A_{\alpha\beta\mu}\biggr{)},$
(36)

and with the aid of the identity
$\partial_{\mu}f_{\nu}dx^{\nu}-df_{\mu}=i_{\partial_{\mu}}\mbox{rot}(f_{\nu}dx^{\nu})$
for a 1-form $f_{\mu}dx^{\mu}$ ($i_{\partial_{\mu}}$ stands for the interior
product wirh $\partial_{\mu}$ and
$\displaystyle\mbox{rot}(f_{\mu}dx^{\mu}):=\frac{1}{2}\left(\frac{\partial{f_{\nu}}}{\partial{x^{\mu}}}-\frac{\partial{f_{\mu}}}{\partial{x^{\nu}}}\right)dx^{\mu}\wedge
dx^{\nu}$ is used for the exterior derivative to differentiate it from the
total derivative $d$), we obtain

$\displaystyle
d\tilde{P}_{a}e^{a}{}_{\mu}-\frac{1}{2}dS^{bc}\omega_{bc\mu}+\frac{1}{2}dJ^{\alpha\beta}A_{\alpha\beta\mu}=\tilde{P}_{a}i_{\partial_{\mu}}\mbox{rot}\theta^{a}-\frac{1}{2}S^{bc}i_{\partial_{\mu}}\mbox{rot}\omega_{bc}+\frac{1}{2}J^{\alpha\beta}i_{\partial_{\mu}}\mbox{rot}A_{\alpha\beta}.$
(37)

Substituting (20), (30) and the definitions of torsion and curvature of the
spacetime and the field strength of the gauge field $A$

$\displaystyle\frac{1}{2}T^{a}{}_{bc}\theta^{b}\wedge\theta^{c}:=\mbox{rot}\theta^{a}+\omega^{a}{}_{b}\wedge\theta^{b}$
(38)
$\displaystyle\frac{1}{2}R_{abcd}\theta^{c}\wedge\theta^{d}:=\mbox{rot}\omega_{ab}+\omega_{a}{}^{c}\wedge\omega_{cb}$
(39) $\displaystyle\frac{1}{2}F_{\alpha\beta
ab}\theta^{a}\wedge\theta^{b}:=\mbox{rot}A_{\alpha\beta}+A_{\alpha}{}^{\gamma}\wedge
A_{\gamma\beta}$ (40)

into the above equation, we finally obtain

$\displaystyle
D\tilde{P}_{a}=\tilde{P}_{b}T^{b}{}_{ac}\theta^{c}-\frac{1}{2}S^{bc}R_{bcad}\theta^{d}+\frac{1}{2}J^{\alpha\beta}F_{\alpha\beta
ab}\theta^{b}.$ (41)

Here, the covariant derivative of the momentum is defined by

$\displaystyle D\tilde{P}_{a}:=d\tilde{P}_{a}-\omega^{b}{}_{a}\tilde{P}_{b}.$
(42)

The second term in the equation (41) is seen in the MPTD equations and the
first term with torsion appears because we are considering this model on a
Riemann-Cartan spacetime, which is consistent with what Leclerc Leclerc  and
Ünal Unal  have derived. The last term represents the field strength and the
color current coupling in Wong equations. The equations (20), (30) and (41)
express a generalization of the MPTD equations and Wong equations.

##  III MPTD equations with Rarita-Schwinger field

###  III.1 Model

In this section, we consider a model which is invariant under simple
supersymmetry. The parameter $\kappa$ and the gauge field $A$ are set to be
zero. The spinor is now expressed as $|\xi\rangle=|_{A}\rangle\xi^{A}$ without
color freedom. Instead, the Rarita-Schwinger field

$\displaystyle|\psi\rangle=|\psi_{\mu}(x)\rangle
dx^{\mu}=|_{A}\rangle\psi^{A}(x)=|_{A}\rangle\psi^{A}{}_{\mu}(x)dx^{\mu}$ (43)

is introduced to preserve the supersymmetry. We consider the following
Lagrangian throughout this section:

$\displaystyle
L=-m\sqrt{\eta_{ab}\Pi^{a}\Pi^{b}},\quad\Pi^{a}=\theta^{a}+\langle\xi|\gamma^{5}\gamma^{a}|D\xi\rangle+\langle\xi|\gamma^{5}\gamma^{a}|\psi\rangle,\quad|D\xi\rangle=|d\xi\rangle+\omega|\xi\rangle.$
(44)

As in section II, several transformations and the conjugate equations are
considered so that the generalized MPTD equations with the Rarita-Schwinger
field.

###  III.2 Vector fields and the equations for their conjugate quantities

####  III.2.1 Supersymmetry

The model (44) is invariant under the super transformation

$\displaystyle\delta\xi^{A}=\epsilon^{A},\quad\delta\theta^{a}=-\langle\epsilon|\gamma^{5}\gamma^{a}|D\xi\rangle-\langle\epsilon|\gamma^{5}\gamma^{a}|\psi\rangle,\quad\delta|\psi\rangle=-|D\epsilon\rangle=-\left(|d\epsilon\rangle+\omega|\epsilon\rangle\right),$
(45)

where $\epsilon^{A}(x)$ are arbitrary Grassmann odd functions of $x^{\mu}$. In
fact, this transformation directly leads to $\delta\Pi^{a}=0$. Unlike the
original Casalbuoni-Brink-Schwarz Lagrangian defined on a flat spacetime, our
superparticle model is supersymmetric even on a Riemann-Cartan spacetime.

Correspondingly, we consider a vector field

$\displaystyle v=\epsilon^{A}\frac{\partial{}}{\partial{\xi^{A}}}$ (46)

for arbitrary Grassmann odd constants $\epsilon^{A}$. The equation (13)
becomes

$\displaystyle\epsilon^{A}\frac{\partial{L}}{\partial{\xi^{A}}}=d\left[\epsilon^{A}\frac{\partial{L}}{\partial{d\xi^{A}}}\right],$
(47)

i.e.,

$\displaystyle\epsilon^{A}P_{a}\biggl{(}-\langle_{A}|\gamma^{5}\gamma^{a}|d\xi\rangle-\langle_{A}|\gamma^{5}\gamma^{a}\omega|\xi\rangle-\langle_{A}|\gamma^{5}\gamma^{a}|\psi\rangle-\langle\xi|\gamma^{5}\gamma^{a}\omega|_{A}\rangle\biggr{)}=\epsilon^{A}dQ_{A}.$
(48)

Here, we take

$\displaystyle Q_{A}:=-P_{a}\langle\xi|\gamma^{5}\gamma^{a}|_{A}\rangle$ (49)

as the supercharge. Defining the covariant derivative of the supercharge by

$\displaystyle
DQ_{A}=dQ_{A}-\frac{1}{4}\omega_{ab}Q_{B}(\gamma^{ab})^{B}{}_{A}=\langle
dQ|_{A}\rangle-\langle Q|\omega|_{A}\rangle,$ (50)

we obtain

$\displaystyle(DQ)_{A}=-P_{a}\left(\langle_{A}|\gamma^{5}\gamma^{a}|D\xi\rangle+\langle_{A}|\gamma^{5}\gamma^{a}|\psi\rangle\right).$
(51)

####  III.2.2 Lorentz transformation

In addition to the transformation (22), we consider

$\displaystyle\delta|\psi\rangle=\frac{1}{4}\epsilon_{ab}\gamma^{ab}|\psi\rangle$
(52)

for the Rarita-Schwinger field. This Lagrangian is also invariant under the
Lorentz transformation. The equation (13) for the vector field

$\displaystyle
v=\frac{1}{4}\epsilon_{ab}(\gamma^{ab})^{A}{}_{B}\xi^{B}\frac{\partial{}}{\partial{\xi^{A}}}$
(53)

with antisymmetric constant $\epsilon_{ab}$ becomes

$\displaystyle\frac{1}{4}\epsilon_{ab}(\gamma^{ab})^{A}{}_{B}\xi^{B}\frac{\partial{L}}{\partial{\xi^{A}}}+\frac{1}{4}\epsilon_{ab}(\gamma^{ab})^{A}{}_{B}d\xi^{B}\frac{\partial{L}}{\partial{d\xi^{A}}}=d\left[\frac{1}{4}\epsilon_{ab}(\gamma^{ab})^{A}{}_{B}\xi^{B}\frac{\partial{L}}{\partial{d\xi^{A}}}\right],$
(54)

i.e.,

$\displaystyle\epsilon_{ab}\biggl{(}P^{a}\langle\xi|\gamma^{5}\gamma^{b}|D\xi\rangle-\omega^{a}{}_{c}S^{bc}-\frac{1}{4}P_{c}\langle\xi|\gamma^{5}\gamma^{cab}|\psi\rangle+\frac{1}{2}P^{a}\langle\xi|\gamma^{5}\gamma^{b}|\psi\rangle\biggr{)}=-\frac{1}{2}\epsilon_{ab}dS^{ab}.$
(55)

With the use of identities

$\displaystyle
P^{a}\theta^{b}-P^{b}\theta^{a}=-P^{a}\langle\xi|\gamma^{5}\gamma^{b}|D\xi\rangle+P^{b}\langle\xi|\gamma^{5}\gamma^{a}|D\xi\rangle-P^{a}\langle\xi|\gamma^{5}\gamma^{b}|\psi\rangle+P^{b}\langle\xi|\gamma^{5}\gamma^{a}|\psi\rangle$
(56)

and

$\displaystyle\langle
Q|\gamma^{ab}|\psi\rangle=-P_{c}\langle\xi|\gamma^{5}\gamma^{cab}|\psi\rangle-P^{a}\langle\xi|\gamma^{5}\gamma^{b}|\psi\rangle+P^{b}\langle\xi|\gamma^{5}\gamma^{a}|\psi\rangle,$
(57)

we have

$\displaystyle DS^{ab}=P^{a}\theta^{b}-P^{b}\theta^{a}-\frac{1}{2}\langle
Q|\gamma^{ab}|\psi\rangle.$ (58)

This can be a generalized MPTD equation with the additional Rarita-Schwinger
term coupled with the supercharge. The Tulczyjew condition

$\displaystyle S^{ab}P_{b}=0$ (59)

is also satisfied.

####  III.2.3 Translation

For translation (34), the equation (13) with respect to this Lagrangian is

$\displaystyle\epsilon^{\mu}\frac{\partial{L}}{\partial{x^{\mu}}}+d\epsilon^{\mu}\frac{\partial{L}}{\partial{dx^{\mu}}}=d\biggl{[}\epsilon^{\mu}\frac{\partial{L}}{\partial{dx^{\mu}}}\biggr{]},$
(60)

i.e.,

$\displaystyle\epsilon^{\mu}\biggl{(}P_{a}\partial_{\mu}e^{a}{}_{\nu}dx^{\nu}-\frac{1}{2}S^{bc}\partial_{\mu}\omega_{bc\nu}dx^{\nu}-\langle
Q|\partial_{\mu}\psi_{\nu}\rangle
dx^{\nu}\biggr{)}=\biggl{(}P_{a}e^{a}{}_{\mu}-\frac{1}{2}S^{bc}\omega_{bc\mu}-\langle
Q|\psi_{\mu}\rangle\biggr{)}.$ (61)

Eliminating $\epsilon^{\mu}$, we have

$\displaystyle dP_{a}e^{a}{}_{\mu}-\frac{1}{2}dS^{bc}\omega_{bc\mu}-\langle
dQ|\psi_{\mu}\rangle=P_{a}i_{\partial_{\mu}}\mbox{rot}\theta^{a}-\frac{1}{2}S^{bc}i_{\partial_{\mu}}\mbox{rot}\omega_{bc}-\langle
Q|i_{\partial_{\mu}}\mbox{rot}\psi\rangle.$ (62)

Substituting (51) and (58), we obtain the other half of our generalized MPTD
equations

$\displaystyle DP_{a}=$ $\displaystyle
P_{b}\tilde{T}^{b}{}_{a}-\frac{1}{2}S^{bc}R_{bcad}\theta^{d}-\langle
Q|i_{\partial_{e_{a}}}\mbox{rot}\psi\rangle,$ (63)

where we use the definition

$\displaystyle DP_{a}:=dP_{a}-\omega^{b}{}_{a}P_{b},$ (64)

and

$\displaystyle\tilde{T}^{b}{}_{a}:=T^{b}{}_{ac}\theta^{c}-\langle\psi_{a}|\gamma^{5}\gamma^{b}|D\xi\rangle-\langle\psi_{a}|\gamma^{5}\gamma^{b}|\psi\rangle.$
(65)

This is the torsion modified by the Rarita-Schwinger field. If we consider the
limit $\xi^{A}\to 0$, this modified torsion has the same expression as the one
that appears in the context of supergravity.

##  IV Discussion

In this paper, we have derived the MPTD equations with torsion from a
pseudoclassical Lagrangian of Majorana spinors on a supermanifold. Its
Grassmann even part forms a Riemann-Cartan spacetime, which induces the
torsion term to the original MPTD equations. We have also considered two types
of generalization. The first one has additional internal degrees of freedom
and renormalized mass. This adds the color current term that appeared in the
Wong equations to our equations. For the other generalized model, we introduce
the Rarita-Schwinger field to retain supersymmetry. This modifies the torsion
consistently with supergravity.

Since our second model is a relatively straightforward supersymmetric
extension of the particle Lagrangian of general relativity, it can be
inherently related to supergravity. Its geometric quantities such as
connection and curvature on a supermanifold should be calculated for more
concrete analysis. They will be treated in terms of Finsler geometry.

## References

  * (1) M. Mathisson, Republication of: New mechanics of material systems, Gen Relativ Gravit, 42, 1011 (2010)
  * (2) A. Papapetrou, Spinning Test-Particles in General Relativity. I, Proc. R. Soc. Lond. A, Mathematical and Physical Sciences, 209, 248 (1951)
  * (3) W. Tulczyjew, Motion of Multipole Particles in General Relativity Theory, Acta Phys. Pol. 18, 393 (1959)
  * (4) W.G. Dixon, A Covariant Multipole Formalism for Extended Test Bodies in General Relativity, Nuovo Cimento 34, 317 (1964)
  * (5) D. Puetzfeld, C. Lämmerzahl and B. Schutz, eds., Equations of motion in relativistic gravity, Springer International Publishing (2015)
  * (6) M. Leclerc, Mathisson-Papapetrou equations in metric and gauge theories of gravity in a Lagrangian formulation, Class. Quantum Grav. 22, 3203 (2005)
  * (7) R. R. Lompay, Deriving Mathisson-Papapetrou equations from relativistic pseudomechanics, arXiv:0503054 [gr-qc] (2005)
  * (8) N. Ünal, Electromagnetic and gravitational interactions of the spinning particle, J. Math. Phys. 47, 092501 (2006)
  * (9) W. G. Ramírez and A. A. Deriglazov, Lagrangian formulation for Mathisson-Papapetrou-Tulczyjew-Dixon equations, Phys. Rev. D 92, 124017 (2015)
  * (10) A. Barducci, R. Casalbuoni and L. Lusanna, Classical spinning particles interacting with external gravitational fields, Nucl. Phys. B, 124, 521 (1977)
  * (11) R. Casalbuoni, Relatively and supersymmetries, Phys. Lett., 62B, 49 (1976)
  * (12) L. Brink and J. H. Schwarz, Quantum superspace, Phys. Lett., 100B, 310 (1981)
  * (13) T. Ootsuka, M. Ishida, E. Tanaka and R. Yahagi, Super Finsler Connection of Superparticle on Two Dimensional Curved Spacetime, IJGMMP, 16,1950055 (2019)
  * (14) C. Chevalley, The Algebraic Theory of Spinors and Clifford Algebras, Springer (1996)
  * (15) Y. Suzuki, On the Elementary Theory of Spinor, Journal of the College of Arts and Sciences, Chiba University, 5, 27 (1967)
  * (16) T. Ootsuka, R. Yahagi and M. Ishida, Killing Symmetry on the Finsler Manifold, Class. Quantum Grav. 34, 095002 (2017)


11institutetext: Institute of Astronomy, Faculty of Physics, Astronomy and
Informatics, Nicolaus Copernicus University, Grudziadzka 5, 87-100 Torun,
Poland, 11email<EMAIL_ADDRESS>22institutetext: INAF, Osservatorio
Astronomico di Cagliari, via della Scienza 5, 09047, Selargius, Italy
33institutetext: INAF, Osservatorio Astrofisico di Arcetri, Largo E. Fermi 5,
50125, Firenze, Italy 44institutetext: Joint Institute for VLBI ERIC (JIVE),
Oude Hoogeveensedijk 4, 7991 PD Dwingeloo, The Netherlands 55institutetext:
Sterrewacht Leiden, Leiden University, Postbus 9513, 2300 RA Leiden, The
Netherlands 66institutetext: University of New Mexico, Department of Physics
and Astronomy, Albuquerque, NM 87131, USA

# Proper motion study of the 6.7 GHz methanol maser rings.

I. A sample of sources with little variation

A. Bartkiewicz  11 A. Sanna  22 M. Szymczak  11 L. Moscadelli  33 H.J. van
Langevelde  445566 P. Wolak  11 A. Kobak  11 M. Durjasz  11

(Received 5 February 2024; accepted 28 March 2024)

###### Abstract

Context. Methanol masers at 6.7 GHz are well-known signposts of high-mass
star-forming regions. Due to their high brightness, they enable us to derive
the three-dimensional gas kinematics near protostars and young stars.

Aims. We aim to understand the origin of the ring-like structures outlined by
methanol maser emission in a number of sources. This emission could be, a
priori, spatially associated with an outflow and/or disc around a high-mass
protostar. In cases of expansion or rotation, maser proper motions should be,
for instance, diverging from the ring centre or perpendicular to the ring
radius.

Methods. Using sensitive, three-epoch observations spanning over eight years
with the European VLBI Network, we have started the most direct investigations
of maser rings using very accurate proper motion measurements with
uncertainties below $\sim$1 km s-1.

Results. We present full results for the five targets of our sample,
G23.207$-$00.377, G23.389$+$00.185, G28.817$+$00.365, G31.047$+$00.356, and
G31.581$+$00.077, where proper motions show similar characteristics; maser
cloudlets do not move inwards towards the centre of the rings but rather
outwards. We also include the most circular source, G23.657$-$00.127, in the
discussion as a reference. The magnitude of maser proper motions ranges from a
maximum of about 13 km s-1 to 0.5 km s-1. In two of the five sources with a
high number of maser spots ($>$ 100), namely G23.207$-$00.377 and
G23.389$+$00.185, we show that the size of the best elliptical model, fitted
to the distribution of persistent masers, increases in time in a manner
similar to the case of G23.657$-$00.127. Moreover, we checked the separations
between the pairs of spots from distinct regions, and we were able to assess
that G28.817$+$00.365 and G31.047$+$00.356 can be interpreted as showing
expanding motions. We analysed the profiles of single maser cloudlets and
studied their variability. Contrary to single-dish studies, the
interferometric data indicate variability of the emission of single-masing
cloudlets.

Conclusions. In five of the six targets, namely, G23.207$-$00.377,
G23.389$+$00.185, G23.657$-$00.127, G28.817$+$00.365, and G31.047$+$00.356,
expansion motions prevail. Only in the case of G31.581$+$00.077 can a scenario
of disc-like rotation not be excluded. Complementary observations of thermal
tracers as well as searching for ultra-compact H II regions in the same
sources are needed. Although the overall morphology of the maser emission has
remained stable, the intensities of individual maser cloudlets varied from
epoch to epoch, suggesting internal instabilities.

###### Key Words.:

masers – stars: massive – instrumentation: interferometers – stars: formation
– astrometry

##  1 Introduction

Young high-mass stars with masses larger than 8 M⊙ form inside star-forming
regions, and they are still challenging targets in modern astrophysics, as
they are distant (distances of a few kiloparsecs), obscured at optical and
near-infrared wavelengths, and evolve on short timescales. These massive young
stellar objects (MYSOs) can be studied by means of 6.7 GHz methanol maser
observations (Menten, 1991) because this line has been found to uniquely trace
their environment (e.g. Breen et al. 2013 and references therein). Very long
baseline interferometry (VLBI) allows us to derive the detailed spatial
structure of methanol maser emission at milliarcsecond resolution (e.g.
Moscadelli et al. 2011a, 2011b, Nakamura et al. 2023) and investigate the
small (a few astronomical units) regions of neutral gas in the vicinity of
MYSOs (at radii of 1000s AU). Since the first VLBI observations of the 6.7 GHz
methanol maser emission almost three decades ago, maser spots have been shown
to be found in diverse structures. These structures can consist of a few or
many maser spots, having either linear or arched spatial distributions, which
often show regular velocity gradients as well (e.g. Norris et al. 1993,
Phillips et al. 1998, Walsh et al. 1998, Minier et al. 2000). With the
increase of the VLBI sensitivity, ring-like structures were also discovered
(Bartkiewicz et al. 2009, Torstensson et al. 2011, Fujisawa et al. 2014).
However, the general question about where the methanol maser emission arises
has not yet been answered. Recently, proper motion studies using VLBI data
have added further information to address this question. Such observations
provide the most direct measurements of the gas kinematics near MYSOs,
although long time baselines are required – typically of a few years – to
trace shifts on the order of a few milliarcseconds for a single maser spot.

A series of works by Moscadelli et al. (2000, 2005, 2007, 2011a) demonstrated
the ability of water masers to trace 3D velocities of gas, which usually
expands from MYSOs at several 10 km s-1. Towards two MYSOs, G16.59$-$0.05 and
G23.01$-$0.41, Sanna et al. (2010a, b) also reported accurate multi-epoch
proper motion measurements of the 6.7 GHz masers for the first time, using a
time baseline of two years. This maser emission arises close to the central
protostar (within 1000 AU) and traces gas either rotating and/or expanding
with mean (transverse) velocities of 6–7 km s-1. Similarly, Moscadelli et al.
(2011b) showed that the methanol masers in IRAS 20126$+$4104 are associated
with a rotating disc while also moving in the direction of the jet axis. A
special case was further discovered in AFGL 5142, where the methanol masers
provided a direct measurement of gas infall within a radius of 300 AU from an
intermediate-mass protostar (Goddi et al. 2011). In Cepheus A HW2, the
velocity field is dominated by an infall component of about 2 km s-1 down to a
radius of 300 AU, where a rotational component of 4 km s-1 becomes dominant,
as revealed by the 6.7 GHz methanol maser studies (Torstensson et al. 2011,
Sugiyama et al. 2014, Sanna et al. 2017).

The following project aims to measure the proper motions of those 6.7 GHz
methanol masers distributed along ring-like structures. We selected twelve
objects from our sample of 31 sources that were observed from 2004 to 2007 by
Bartkiewicz et al. (2009) using the European VLBI Network111The European VLBI
Network is a joint facility of independent European, African, Asian, and North
American radio astronomy institutes. Scientific results from data presented in
this publication are derived from the following EVN project codes: EN003,
EB031, EB034, EB052. (EVN) and repeated the observations with a similar setup
in 2013 and 2015. Since 6.7 GHz methanol masers typically trace gas velocities
of less than 10 km s-1 and the targets in our sample have distances within
about 4 kpc from the Sun, one would expect to detect position shifts of 3 mas
after six years. These angular shifts are about half the EVN beam size and are
easily detectable when there is a sufficient signal-to-noise ratio (S/N).

In this paper, we present proper motion results for five of the 12 sources in
our sample. The first sub-sample concerns targets whose ring-like morphology
was stable and where we identified persistent spots distributed along the
whole structure over about eight years. Single-dish monitoring showed little
or no variation of the 6.7 GHz line emission towards the same targets
(Szymczak et al., 2018). Another source, G23.657-00.127, showing a nearly
circular morphology, also belongs to this sub-sample but was analysed and
described in detail in Bartkiewicz et al. (2020). Here, we also include some
results of G23.657$-$00.127 for completeness.

##  2 Observations and data reduction

The observations of the 6668.519 MHz methanol maser line taken from 2004 to
2007 are described in Bartkiewicz et al. (2009) and are referred to here as
epoch 1 (hereafter E1). The observations for epoch 2 and epoch 3 (hereafter E2
and E3, respectively) were carried out using the EVN on March 2013 and 2015
(Table 1). Data were obtained from the following antennas: Jodrell Bank,
Effelsberg, Medicina, Onsala, Noto, Torun, Westerbork, and Yebes. Details of
the observations, such as pointing positions of the targets, phase
calibrators, and exact dates of observations for the five masers, are given in
Table 1. Each observing run lasted for 9 hours. The phase-referencing mode was
used, repeating a loop of 5 min between the maser target (3.25 min) and a
phase calibrator from the VLBA Calibrator List (1.75 min). The calibrator
3C345 was observed during a few 15-min scans at each run; these scans were
used to calibrate the bandpass and instrumental phases. Three maser sources
were observed in each 9-hr run; these targets were selected to be within a few
degrees of each other, in projection on the sky, and of similar maser emission
velocities (Table 1). This process yielded the on-source time for each maser
of ca. 1.5 hr. The SFXC software correlator in JIVE was used with a 2-second
integration time (Keimpema et al., 2015). Selection of 1024 channels in the 2
MHz bandwidth resulted in a 0.1 km s-1 resolution. To increase S/N on the
phase-reference source, eight 2-MHz BBCs per polarisation were used for a
second correlator pass with 128 channels per BBC.

The data reduction was carried out with the Astronomical Image Processing
System (AIPS) developed by the National Radio Astronomy Observatory (NRAO)
with standard procedures for spectral line observations. The Effelsberg
antenna was set as the reference. As we aimed for detailed internal motion
studies of masers and relied on the relative motions, fringe fitting (task
FRING) was performed on the strongest maser channel from the first-epoch
observations (Bartkiewicz et al., 2009). This allowed us to exclude motions
due to annual parallax and Galactic rotation and also provided the highest
possible S/N to get the most accurate measurements of the relative positions.
To obtain the positions of the methanol maser spots in all channel maps, we
used the AIPS tasks JMFIT or SAD (depending on the number of spots per source)
fitting two-dimensional Gaussian models to the compact maser emission. The
formal fitting errors were typically less than 0.1 mas per spot. The rms noise
level (1$\sigma_{\rm rms}$) in line-free channels was typically 4 or 6 mJy for
each source, and we searched for maser emission above a threshold of
7$\sigma_{\rm rms}$ in each channel map. The synthesised beams, obtained with
natural weightings in E2, are also listed in Table 1.

Table 1: Details of EVN observations.

Sourcea | Pointing positions (J2000) | Velocity | Phase-calibrator | Observing | Synthesised beamc
---|---|---|---|---|---
|  |  | peak |  | runsb (and 1$\sigma_{\rm rms}$ levels) |
Gll.lll$\pm$bb.bbb | RA (h:m:s) | Dec (o:’:”) | $({\rm km\leavevmode\nobreak\ s}^{-1})$ |  | (mJy) | (mas$\times$mas; PA(°))
G23.207$-$00.377 | 18:34:55.21212 |  $-$08:49:11.8926 | 77.1 | J1825$-$0737 | A(4), E(4) | 8.3$\times$4.1; $-$27
G23.389$+$00.185 | 18:33:14.32477 |  $-$08:23:57.4723 | 75.4 | J1825$-$0737 | A(4), E(4) | 8.3$\times$4.1; $-$27
G23.657$-$00.127 | 18:34:51.56482 |  $-$08:18:21.3045 | 82.6 | J1825$-$0737 | A(4), E(4) | 9.0$\times$4.0; $-$28
G28.817$+$00.365 | 18:42:37.34797 |  $-$03:29:40.9216 | 90.7 | J1834$-$0301 | C(4), G(4) | 7.9$\times$4.1; $-$30
G31.047$+$00.356 | 18:46:43.85506 |  $-$01:30:54.1551 | 80.7 | J1834$-$0301 | C(4), G(4) | 7.5$\times$4.1; $-$32
G31.581$+$00.077 | 18:48:41.94108 |  $-$01:10:02.5281 | 95.6 | J1834$-$0301 | D(6), H(6) | 9.3$\times$5.5; $+$4.7

222a The names are the Galactic coordinates of the brightest maser spots as in
Bartkiewicz et al. (2009). b The project EB052 was observed in the following
runs: A (2/03/2013), C (4/03/2013), D (5/03/2013), E (15/03/2015), G
(18/03/2015) and H (19/03/2015). c The sizes as in the runs A, C, D carried
out in 2013.

###  2.1 Proper motions measurements

In order to study the proper motion of the masers, we followed the general
procedure (method 1): i) First we determined the maser ‘cloudlets’ at each
epoch, where a maser cloudlet is defined as a combination of at least three
(in rare cases two) maser spots appearing at consecutive spectral channels and
coinciding in position within half the synthesised beam (e.g. Sanna et al.
2017). ii) Next, we identified the persistent maser cloudlets over the epochs
on the basis of their linear motion with respect to a bright, compact, and
spectrally stable feature. Relative proper motions were derived via linear
fits of cloudlet displacements over the three epochs. iii) Last, we subtracted
the average proper motion of persistent cloudlets from the proper motion of
each cloudlet. As a result of this, we defined a centre of motion
approximating the MYSO’s rest frame, with respect to which we calculated the
proper motions. This is equivalent to calculating the geometric mean of the
positions of all persistent maser cloudlets at each epoch and then referring
the proper motions to this point. We have already discussed this approach in
Bartkiewicz et al. (2020) for G23.657$-$00.127.

For two targets with a significant number of spots ($>$100), namely
G23.207$-$00.377 and G23.389$+$00.185 (as well as in G23.657$-$00.127 as
presented in Bartkiewicz et al. (2020)), we also followed a second approach
for the proper motion analysis, taking advantage of the best-fitted ellipses
to the maser distribution. This second approach (method 2) involved four
steps: i) selection of maser spots that were visible at all three epochs; ii)
fitting the flux-weighted ellipses to the selected maser spots at each epoch
using the code by Fitzgibbon et al. (1999); iii) aligning of the centres of
best-fitted ellipses with respect to the first epoch; iv) constructing the
averaged proper motion vector (i.e. the displacements between pairs of spots
within a group were summed and divided by the number of pairs for each group
of maser spots that were clearly separated from each other). Assuming that the
methanol emission traces an ellipse and that the centre of the ellipse
coincides with the MYSO position enabled us to remove any bulk motion of the
ring in the plane of the sky and obtain the proper motions of masers relative
to MYSO.

As mentioned earlier, the formal fitting errors of maser spot positions were
typically negligible, less than 0.1 mas. This evidence strengthens the
accuracy of the proper motion measurements. Also, the long time baseline (8-11
yr) of the observations is an advantage of the present study, considering that
the targets are at distances of a few kiloparsecs. We assumed in our studies
that co-spatial maser spots at the same LSR velocity identified at different
epochs trace the same volume of gas. In method 1, we derived the proper
motions via linear fits of cloudlet displacements over three epochs. We drew
‘error cones’ for each proper motion vector, as derived from the uncertainty
of the linear fits in each direction (RA and Dec). In method 2, we derived the
averaged proper motion vectors between E1-E2 and E1-E3 data and drew error
cones as a range of these two directions.

###  2.2 Single-dish monitoring

The whole sample has been monitored with the 32-m Torun radio telescope under
the project described in Szymczak et al. (2018) in the period 2009-2013. In
addition, follow-up observations have been ongoing since early 2018. The
spectral resolution was 0.09 km s-1 after Hanning smoothing, a typical
sensitivity (3$\sigma$) was 0.8 Jy, and the flux density calibration accuracy
was about 10%.

##  3 Results

In this section, we present the results for five sources that show similar
characteristics in their proper motions and for which the single-dish
monitoring showed little or no variation of the 6.7 GHz line Szymczak et al.
(2018). The archetype ring-like source G23.657$-$00.127 is presented in detail
in Bartkiewicz et al. (2020), and here we only report some characteristics for
comparison.

In Fig. 7 we present the distribution of the 6.7 GHz methanol maser emission
(all visible spots at single maser channel maps) at all three epochs of
observations. Also, we present the spectra of the cloudlets and analyse their
Gaussian profiles. The dynamic spectra of the 6.7 GHz methanol maser
variability, obtained using the Torun dish, are presented in Fig. 8. Distances
to the targets, derived either from trigonometric parallax measurements or
kinematic distances, have been added to Table 2, where the flux-weighted
ellipse fits to the distribution of the persisting cloudlets are summarised.
The maser cloudlets with derived proper motions are listed in Table
LABEL:clouds and presented in Figs 1-5. One can note that the overall
structures in all cases persisted over almost eight years (G28.817$+$00.365,
G31.047$+$00.356) or even 10.5 years (G23.207$-$00.377, G23.389$+$00.185,
G23.657$-$00.127). In general, displacements of the cloudlets suggest outward
motions from the centres of the rings (i.e. fitted ellipses).

Figure 1: Proper motions of 6.7 GHz methanol maser cloudlets in
G23.207$-$00.377. Top: Proper motions as measured relative to the centre of
motion. The black arrows indicate the best fits of the relative proper motion
over the three-epoch data, and the uncertainties are marked by the grey
triangles. The centre of motion is marked by the plus sign and explained in
Sect. 2.1 (method 1); the (0,0) point is offset by ($-$62.59255 mas, 50.5022
mas) relative to the brightest spots listed in Table 1. The circle sizes are
proportional to the logarithm of the flux of the brightest maser spots of each
cloudlet at epoch E1, and the colours relate to their LSR velocities as shown
on the vertical wedge. The dotted ellipse traces the best flux-weighted fit to
all cloudlets as detected in E1 (Table 2), and its centre is marked with a
cross. Bottom: Proper motions estimated using method 2 by fitting the ellipse
at each epoch and aligning their centres. The (0,0) point, marked by a cross,
corresponds to the centre of the best-fitted ellipse in E1 (Table 2). The
black arrows represent the averaged proper motion vectors between E1-E2 and
E1-E3 data.

For G23.207$-$00.377, the following number of spots were registered during
observations in 2004, 2013, and 2015: 218, 219, and 250, respectively (Fig.
7). We identified 140 spots that appeared in all three epochs and that can be
grouped into 23 cloudlets. These cloudlets show Gaussian profiles (Fig. 7)
with varying flux densities for most of them between 2004 and 2013 (Table
LABEL:clouds). The brightest cloudlet in 2004, at the LSR velocity of 77.1 km
s-1, decreased by a factor of more than four after 8.5 yr. This is contrary to
the results of single-dish monitoring, where the overall emission from this
target in recent years (since 2009) shows little spectral variation (Fig. 8;
see also Szymczak et al. (2018)). In the poor signal-to-noise ratio
observations and without any possibility to spatially resolve spectrally
blended features, it is impossible to detect single-cloudlet variability. As
we stated in Bartkiewicz et al. (2020), the internal instabilities of the
masing regions may reduce the velocity coherence along the path of maser rays
and induce flux variations. However, we also note that the difference in the
variability patterns from VLBI and the 32-m single-dish monitoring possibly
results from the blending effect, as VLBI probes the most compact cloud or the
core of the cloudlet.

We selected 20 cloudlets with approximately linear motions over three epochs
(with respect to the brightest one) and used them for calculating the average
proper motion. The remaining three cloudlets are marked in Table LABEL:clouds.
Figure 1 presents the derived proper motions for G23.207$-$00.377. This plot
highlights the fact that the majority of cloudlets move outwards from the
centroid of their distribution. Their motions range from 0.06 to 0.58 mas
yr-1, which at a distance of 4.18 kpc (Reid et al., 2019) correspond to linear
velocities from 1.2 to 11.6 km s-1. We calculated the radial and tangential
components of the proper motions (Fig. 9), and there is a clear outward radial
motion in the sky with a mean value of 2.61$\pm$0.09 km s-1. The tangential
components do not show any consistent rotation pattern. Following method 2 to
measure maser proper motions, by aligning the centre of the ellipses at
different epochs, we obtained results consistent with method 1 after excluding
the blueshifted spots to the SW of the ellipse fit. The major axis of the
best-fitted ellipses increased in time, with a linear size of 1072, 1082, and
1084 AU in 2004, 2013, and 2015, respectively (Table 2), indicating the same
outward motion of the masing cloudlets.

Figure 2: Same as Fig. 1 but for G23.389$+$00.185. The centre of motion
(marked by the plus sign) from method 1, the (0,0) point in the left panel, is
shifted by ($-$16.8016 mas, $-$73.825 mas) relative to the brightest spots
listed in Table 1. The (0,0) point in the right panel is defined by the centre
of the best-fitted ellipse in E1 (Table 2).

For G23.389$+$00.185, 128, 118, and 129 maser spots were found in 2004, 2013,
and 2015, respectively. Their elliptical fit shows an elongation towards the
NE-SW direction (Fig. 7). Seventy-nine spots appeared at all three epochs and
are grouped in ten cloudlets. They all showed linear motions and were used to
derive the average proper motion relative to the brightest cloudlet at the LSR
velocity of 75.338 km s-1. The result is presented in Fig. 2. Proper motions
range from 0.02 to 0.7 mas yr-1, corresponding to 0.5 to 16.1 km s-1 at a
distance of 4.8 kpc (Reid et al., 2019). Between method 1 and method 2, the
results are consistent. The major axis of the best-fitted ellipses increased
in time by ca. 30 mas over 10.5 yrs. The radial and tangential components of
proper motions calculated in method 1 are presented in Fig. 9. Again, the
outward radial motion dominates, and its mean is 3.4$\pm$0.4 km s-1. This
source shows little variation in the 6.7 GHz methanol maser emission in recent
years (Fig. 8), as reported by Szymczak et al. (2018). However, we detected an
overall decrease in the flux density of the brightest spectral features,
particularly between 2004 and 2013, and also of the redshifted spectral
features (Fig. 7). Moreover, the source has recently undergone an outburst,
reported by Tanabe & Yonekura (2023). The spectral features at the LSR
velocities of 74.7 and 75.3 km s-1 have increased their flux densities to 100
and 120 Jy, respectively. Interferometric observations are needed to properly
identify the outbursting region.

In our sample, G23.657$-$00.127 is the target with the most circular and
widespread distribution of methanol maser spots. Details were described in
Bartkiewicz et al. (2020), and we summarise them here for completeness of the
sub-sample. Radially outward motions were clear, with values from 0.5 to 5.4
km s-1 in the sky. Consistent results of radially outward motions were
obtained by applying either method 1 or method 2 when we excluded the
blueshifted cloudlet to the SW for the ellipse fit. The major axis of the
best-fitted ellipses increased about 7 mas over 10.5 yr (Table 2). Again, the
single-dish monitoring of the 6.7 GHz methanol maser spectrum showed non-
variable emission over nearly 20 yr within an accuracy of 10%. (Szymczak et
al., 2018). However, our EVN observations revealed significant changes in the
intensity of the individual cloudlets over the whole range of velocity.

Figure 3: Same as Fig. 1 but for G28.817$+$00.365. Due to the significantly
smaller number of cloudlets, only method 1 was used for the proper motion
studies. The centre of motion is marked by the plus sign, the (0,0) point, and
it is shifted by ($-$30.0824 mas, $-$33.9852 mas) relative to the brightest
spots listed in Table 1.

For G28.817$+$00.365, 36, 67, and 80 maser spots were detected in 2007, 2013,
and 2015, respectively (Fig. 7). Twenty-one of them appeared at all three
epochs and are grouped in seven cloudlets. We used five of them that show
approximately linear motions to derive the average proper motion (the excluded
cloudlets are marked in Table LABEL:clouds). The proper motions are presented
in Fig. 3. They range from 0.07 to 0.6 mas yr -1, corresponding to a range
from 1.6 to 13.0 km s-1 for a kinematic distance of 4.6 kpc (Reid et al.,
2019). The decomposition of the proper motion velocity vectors into radial and
tangential components is presented in Fig. 9. Due to the small number of
cloudlets, the domination of any of these cannot be estimated. The mean radial
motion is 3.5$\pm$0.4 km s-1. Gaussian profiles of maser cloudlets show that
the peak of the brightest one (at the LSR velocity of 90.6 km s-1) decreased
in time by a factor of three between 2007 and 2013. More complex spectral
profiles also appeared in the LSR velocity range 91–92 km s-1 between 2007 and
2013. Again, this is opposite to the single-dish monitoring results (Fig. 8;
Szymczak et al. 2018).

Figure 4: Same as Fig. 3 but for G31.047$+$00.356. We note that all maser
spots (instead of cloudlets, as in the other targets) are presented due to the
complex structure in the western region, which is also zoomed-in for clarity
in the left panel. The centre of motion is marked by the plus sign, the (0,0)
point, and it is shifted by ($-$7.9729 mas, $-$0.6561 mas) relative to the
brightest spots listed in Table 1.

Towards G31.047$+$00.356, 31, 77, and 83 single maser spots were detected in
2007, 2013, and 2015, respectively (Fig. 7). Twenty-three spots appeared at
all three epochs and are grouped in four cloudlets. Uniquely for this target,
we analysed the motions of all single spots because of the significant
extension of the western maser emission that would be missed when grouping
spots into cloudlets. The brightest spot was spectrally stable (at the LSR
velocity of 80.709 km s-1) and was selected as a reference. Twenty-one spots
showed linear motions and were used for calculating the average proper motion
(the two excluded spots are marked in Table LABEL:clouds). Proper motions are
presented in Fig. 4. They range from 0.03 to 0.31 mas yr-1, corresponding to a
range from 0.7 to 7.1 km s-1 for a kinematic distance of 4.9 kpc (Reid et al.,
2019). Similarly, as in G28.817$+$00.365, the decomposition of proper motion
velocity vectors into radial and tangential components does not indicate a
dominant expansion or rotation motions (Fig. 9). Gaussian profiles of
cloudlets between 2007 and 2013 exhibit significant changes (Fig. 7), and we
did not find any non-variable feature. Also, we noticed that the cloudlet at
the LSR velocity of 82.99 km s-1 shows the radial motion component of $-$0.10
m s-1 yr-1. The single-dish monitoring from 2009 did not reveal strong
variability, although there is a systematic spectral drift of the redshifted
components (Fig. 8).

Figure 5: Same as Fig. 3 but for G31.581$+$00.077. The centre of motion marked
by the plus sign, the (0,0) point, is shifted by (71.0853 mas, 48.6932 mas)
relative to the brightest spots listed in Table 1.

Towards the source G31.581$+$00.077, 35, 43, and 49 single maser spots were
detected in 2007, 2013, and 2015, respectively (Fig. 7). Nineteen of them were
identified as appearing at all three epochs. They were grouped in eight
cloudlets. Six of them showed linear motions and were used to derivate average
motions. Proper motions are presented in Fig. 5, and the decomposition of them
into radial and tangential components is presented in Fig. 9. We did not
notice any sign of regularities in them. This source lies at 5.5 kpc, as its
trigonometric parallax was measured by Reid et al. (2019). Again,
interferometric data show amplitude variability of the spectral features,
contrary to the single-dish data (Fig. 7).

At this point we can state that the differences in variability patterns based
on VLBI and single-dish observations likely result from significant
differences in angular resolution (arcmin vs. mas). Detailed studies of radial
velocity drifts of single cloudlets combined with the proper motions may
uniquely allow the 3D velocities to be estimated.

Table 2: Parameters of flux-weighted ellipses fitted to the maser cloudlets
that appeared in all three epochs. First-epoch fits are presented in Figs 1-5.

Sourcea | Centrea | Semi-axes; PAmajorb
---|---|---
Distance |  $\Delta$RA |  $\Delta$Dec |
(kpc) | (mas) | (mas) | (mas$\times$mas; °)
G23.207$-$00.377 |  $-$55.817 |  $+$81.207 | 128.33$\times$40.07; $-$48
4.18c |  $-$56.378 |  $+$82.097 | 129.48$\times$40.80; $-$48
|  $-$56.573 |  $+$82.302 | 129.74$\times$40.90; $-$48
G23.389$+$00.185 |  $-$39.045 |  $-$74.610 | 101.84$\times$51.41, $+$52
4.8c |  $-$40.010 |  $-$75.244 | 105.16$\times$51.06; $+$52
|  $-$40.023 |  $-$75.514 | 105.10$\times$51.18; $+$52
G23.657$-$00.127 |  $-$66.058 |  $-$96.411 | 134.89$\times$123.94; $-$20
3.19d |  $-$66.379 |  $-$96.246 | 136.01$\times$124.49; $-$19
|  $-$66.824 |  $-$97.350 | 136.06$\times$125.27; $-$19
G28.817$+$00.365 |  $-$47.443 |  $-$14.272 | 61.53$\times$26.47; $+$53
4.6e |  $-$46.509 |  $-$13.680 | 59.77$\times$27.41; $+$53
|  $-$46.429 |  $-$13.658 | 59.65$\times$27.61; $+$52
G31.047$+$00.356⋆ |  $-$31.068 |  $-$11.355 | 34.28$\times$12.57; $+$63
4.9e |  $-$30.665 |  $-$10.780 | 33.54$\times$14.04; $+$63
|  $-$30.855 |  $-$10.639 | 33.76$\times$14.07; $+$64
G31.581$+$00.077 |  $+$46.437 |  $+$69.870 | 108.22$\times$76.09; $+$87
5.5c |  $+$46.626 |  $+$69.040 | 108.18$\times$74.62; $+$86
|  $+$46.756 |  $+$68.912 | 108.18$\times$74.55; $+$86
333a The relative coordinates to the brightest spot position in each target in
the first epoch as given in Table 1. b The position angle of the major axis
(north to east). Distances were derived either through the trigonometric
parallaxes, c Reid et al. (2019) and d Bartkiewicz et al. (2008), or the
calculated near kinematic distances, e Reid et al. (2019); the online
calculator is available at http://bessel.vlbi-astrometry.org/node/378. ⋆ The
fitting was done to all maser spots visible at each epoch separately (Fig. 7).

##  4 Discussion

###  4.1 Proper motions: Expansion

In the six targets presented here, the overall distributions of methanol maser
emission along ring-like structures have generally been stable for up to eight
to ten years. We detected the variability of individual maser cloudlets,
although the single-dish monitoring suggested non-variable emission overall.
The proper motions of single maser cloudlets have been estimated to range from
a maximum of 13.5 km s-1 to 0.5 km s-1. These values are in agreement with
other studies of proper motions of the 6.7 GHz methanol masers (e.g. Sanna et
al. 2010a). It can be clearly seen, especially in the case of the targets with
a significant number of spots (Figs 1-5), that the proper motions are
consistent with expansion rather than infall, as was claimed for two well-
known MYSOs: Cep A (Torstensson et al. 2011; Sugiyama et al. 2014, Sanna et
al. 2017) and AFLG 5142 (Goddi et al., 2011).

The best elliptical fits to the maser distribution in G23.207$-$00.377,
G23.389$+$00.185, and G23.657$-$00.127 indicate that the sizes of their major
axes have increased in time (Table 2). This result is consistent with our
independent measurement of the maser proper motions (method 1 in Sect. 2.1),
providing support to the idea that the maser distribution expands from a
common centre and is part of a coherent kinematic structure. We also note that
the displacements of single maser spots lying outside the best-fitted ellipses
are consistent with expansion as well. In the case of the remaining three
targets, where a smaller number of maser spots (and cloudlets) were detected,
the increase of ellipse axes is not obvious. Also, the proper motion studies
using the method 1 (Sect. 2.1) do not imply clear expansions in these cases.

Figure 6: Histograms of separations between SE and NW maser spot pairs in each
epoch and the separation increase between epochs E2 and E3 relative to E1 in
G23.207$-$00.377. The dashed lines mark the mean histogram values, while the
dotted lines mark the median values. The histograms for the remaining targets
are presented in Fig. 13.

In order to quantify the internal motion of maser spots in each source in a
manner independent from the methods described above, we calculated the
separations between pairs of spots from distinct regions at each epoch. In
Figs 6 and 13, we present histograms of separations between given groups (the
separated ones) of cloudlets in each epoch. For clarity, we also present the
changes in these separations, that is, the differences between epochs: i) 2004
(or 2007) and 2013, and ii) 2004 (or 2007) and 2015. The mean and median
values are also marked with dashed and dotted lines, respectively.

In the case of G23.207$-$00.377, we used 72 blueshifted maser spots from the
SE part and 40 spots from the NW (selected at the intermediate LSR
velocities). The mean and median values of the spots’ separations increased by
2.31 and 2.95 mas, respectively, between 2004 and 2013, and by 2.95 and 3.09
mas between 2004 and 2015. An increase in the spots’ separation of 3 mas
corresponds to velocities of 7.18 km s-1 (for the time baseline 2004-2013) and
5.77 km s-1 (for 2004-2015) at the distance of the source (Table 2).

In G23.389$+$00.185, we calculated the separation between pairs of spots from
the NE (32 spots) and the SW parts (27 spots). The mean and median values also
increased (2.5 and 1.6 mas for separations between 2004 and 2013, and 2.9 and
2.0 mas between 2004 and 2015, respectively). For this source, an increase in
the spots’ separation of 2 mas corresponds to 5.5 km s-1 (2004-2013) and 4.4
km s-1 (2004-2015).

In G23.657$-$00.127, we considered pairs of spots lying to the north and south
of the EW line at $\Delta$Dec$=20$ mas (see Fig. 1 in Bartkiewicz et al.
(2020)). The northern group contained 88 spots, and the southern one had 120
spots. The mean and median values of their relative separation increased by
1.66 and 1.70 mas for the time baseline 2004-2013 and by 2.13 and 2.15 mas
between 2004 and 2015. At the distance of the source, an increase of 2 mas
corresponds to velocities of 3.7 km s-1 (2004-2013) and 2.9 km s-1
(2004-2015).

In G28.817$+$00.365, we considered 19 spots located in the eastern part and
two spots from the western part. The mean (median) values of changes in
separations are 0.82 and 0.72 mas (0.86 and 0.63 mas) for a time baseline of
six and eight years, respectively. An increase in the spots’ separation of 1
mas from 2007 to 2013 or from 2007 to 2015 corresponds to velocities of 4.0 km
s-1 and 2.9 km s-1, respectively.

Similarly, in 31.047$+$00.356 we calculated the separations between pairs of
spots at east and west, and their mean and median values of changes in
separations are as follows: 0.64 and 1.44 mas (2007-2013) and 1.08 and 2.08
mas (2007-2015), respectively. Here, a displacement of 1 mas from 2007 to 2013
or from 2007 to 2015, corresponds to velocities of 3.8 km s-1 and 2.8 km s-1,
respectively.

Finally, in G31.581$+$00.077, we calculated the separations between spots from
the east (seven) and west (nine spots) regions. The means of their changes are
$-$0.11 and $-$0.03 mas, and the median values are 0.02 and $-$0.01 mas for
2007-2013 and 2007-2015, respectively.

The separation analysis confirmed that for sources G23.207$-$00.377,
G23.389$+$00.185, and G23.657$-$00.127, a clear overall expansion is detected.
In addition, a tentative expansion is seen in G28.817$+$00.365 and
G31.047$+$00.356. For these two latter sources, only a small number of spots
was available, and the time baseline of the observations was shorter, so the
results are less robust and need to be confirmed with future observations. In
the case of G31.581$+$00.077, which is the most distant target from our
sample, we cannot exclude a sign of rotation considering the separation
analysis and the proper motion studies. The fitted ellipses have similar axes
in three epochs of observations as well. Therefore, this scenario has to be
confirmed by future observations and repeated proper motion studies with a
longer time baseline.

###  4.2 Infrared and radio counterparts

In general, it is difficult to relate the 6.7 GHz maser spatial morphology,
which we measured at the mas scale, to the position of the young star, which
can be inferred, for instance, from infrared and radio continuum observations
at scales more than an order of magnitude larger so far. In the following, we
collect the information available in the literature for the five sources.

Inspection of mid-infrared data from the Spitzer IRAC maps, GLIMPSE, and
MIPSGAL444https://irsa.ipac.caltech.edu/ revealed that three out of the six
targets (i.e. G23.207$-$00.377, G23.389$+$00.185, and G23.657$-$00.127)
coincide with unresolved sources. They are also so bright that they saturate
the detectors; the coincidence is within a pixel (i.e.
1$\aas@@fstack{\prime\prime}$2) in a GLIMPSE map. De Buizer et al. (2012)
presented high-resolution near- and mid-infrared images of G23.389$+$00.185
and G23.657$-$00.127. The emission peaks are located northward (ca.
0$\aas@@fstack{\prime\prime}$25) from the methanol maser ring
G23.389$+$00.185. Similarly, the near- and mid-infrared peaks are slightly
shifted southward from the ring G23.657$-$00.127. In both cases, there was no
compelling evidence in support of the hypothesis that methanol masers resided
in circumstellar discs. Hu et al. (2016) also reported weak (0.38 mJy)
continuum emission at 6 GHz towards G23.389$+$00.185, where the emission shows
an elongated morphology that could be interpreted as a jet associated with the
methanol masers. However, the poor astrometry and sensitivity of the radio
continuum observations did not allow for a detailed comparison of position and
morphology with the masers. Furthermore, Bartkiewicz et al. (2009) reported a
non-detection of 8.4 GHz continuum emission towards these three sources, with
an upper limit of 150 $\mu$Jy beam-1.

In contrast, the 6.7 GHz methanol masers in G28.817$+$00.365 were found within
0$\aas@@fstack{\prime\prime}$1 of the position of the 8.4 GHz continuum peak
(Bartkiewicz et al., 2009). This emission, likely an ultra-compact HII region,
was also imaged at 6 GHz by Hu et al. (2016). A search for mid-infrared data
from the catalogues mentioned above did not show any clear counterpart.
Similarly, the G31.047$+$00.356 masers are offset from the mid-infrared
sources. Hu et al. (2016) reported weak continuum emission at 6 GHz that
likely coincides with the methanol emission; this emission was previously
undetected at 8.4 GHz by Bartkiewicz et al. (2009) with an upper limit of 150
$\mu$Jy beam-1. In the case of G31.581$+$00.077, a continuum source was
detected by Hu et al. (2016) but with no evidence supporting the physical
relationship between the maser and continuum source since the maser is offset
eastward by 0.19 pc.

We searched the Atacama Large Millimeter/Submillimeter Array (ALMA) Archive
and found three sources with available data: G23.389$+$00.185,
G23.657$-$00.127, and G31.581$+$00.077. They were part of the large programme
ALMAGal (ALMA Evolutionary study of High Mass Protocluster Formation in the
Galaxy; ID: 2019.1.00195.L). The observations covered the frequency range from
216.8 GHz to 244.14 GHz (band 6), with the best spatial resolution being 0.27”
for G23.389$+$00.185 and G23.657$-$00.127 and 5.57” for G31.581$+$00.077.
Methanol masers in G23.389$+$00.185 overlap with the centres of continuum
emission of C18O and CH3CN emission, proving that the local gas is warm and
that the methanol masers arise from a hot molecular core (due to the nearby
presence of CH3CN emission) and a larger gas condensation (traced by C18O). In
G23.657$-$00.127, however, we observed a westward offset by 0.006 pc, and in
G31.581$+$00.077, we observed a south-west offset by ca. 0.21 pc (Fig. 14). A
detailed analysis of the association between thermal and maser lines at high
angular resolution goes beyond the scope of this study, and it is postponed to
a subsequent dedicated paper. Moreover, the angular resolution of the data is
diverse. It is especially poor for G31.581$+$00.077, and the offset might be
marginal.

###  4.3 Origin of the methanol rings

Sanna et al. (2019) targeted the high-mass star-forming G23.01$-$0.41 with the
ALMA at a resolution of 0$\aas@@fstack{\prime\prime}$2 and directly imaged a
molecular disc around a MYSO in thermal methanol (and methyl cyanide). The
disc rotates and undergoes infall around the central star and drives a
molecular jet arising from the inner disc regions ($<$ 800 AU). The
environment surrounding the young star is also one of the richest (Galactic)
sites of methanol maser emission, where the maser cloudlets are distributed
within a radius of 1500 AU from the central star and, on average, show
expanding motions on the order of 7 km s-1 (Sanna et al. 2015, their Fig. 4).
In five out of the six sources presented in this paper (i.e. all but
G31.581$+$00.077), the methanol maser kinematics show similar properties, with
the maser emission distributed over large areas (300-800 AU) and the proper
motions tracing expansion at velocities of several kilometers per second.
These findings might be consistent with a scenario where the methanol masers
are excited in the inner outflow cavity at the interface with a flare disc. A
similar scenario explains the proper motions of SiO masers in Orion Source I,
the closest MYSO, where the masers arise from a wide-angle bipolar wind
emanating from a rotating edge-on disc (Matthews et al., 2010).

In order to test whether this scenario also applies to the methanol maser
rings, complementary observations of thermal tracers at high angular
resolution ($\sim$0$\aas@@fstack{\prime\prime}$1) are needed, which can be
used to interpret the environment of methanol masers. In this respect, we have
started a sensitive ($\sim\mu$Jy) search of radio continuum emission towards
the methanol maser rings with The Karl G. Jansky Very Large Array. These
observations will be able to detect the ionised gas emission, including from
radio thermal jets or hyper-compact H II regions, excited by any B-type young
star associated with the masers and will allow us to pinpoint the driving
source of the masers at a resolution comparable with the extent of the maser
rings. Moreover, we also expect to explore the origin of maser rings in
combination with the accretion disc or outflow tracers based on locations of
the emission peak of dust and thermal lines, such as CH3CN, SiO, and C18O,
relative to the location of the maser.

For comparison, in G23.657$-$00.127 we proposed two scenarios: a maser ring
related to an outflow or a wide-angle wind at the base of a protostellar jet
tracing a combination of rotation around and expansion along the jet
(Bartkiewicz et al., 2020). Similar decomposition of proper motion vectors for
the remaining targets does not show a clearer scenario. In G23.207$-$00.377
and G23.389$+$00.185, the proper motion characteristics are similar to the
ring, and one may assume the same physically connected structure. However, the
rotational components are not convincing in that sense.

###  4.4 Modelling a plane of gas motion

Sanna et al. (2017) showed that the 3D proper motion vectors of the methanol
maser cloudlets in the HMSFR Cepheus A HW2 move along a plane. The 3D velocity
field of methanol maser cloudlets was known from the proper motions and LSR
velocities with subtraction of the region’s systematic velocity. They showed
that the masers were detected within a radius ranging from 300 to 900 AU from
HW2 tracing infall and rotational components. Due to the lack of complementary
data for our sample of methanol maser rings, we were not able to infer the YSO
position and the associated disc nor the jet orientation in order to give a
more robust interpretation similar to what was presented in Cerpheus A HW2.
However, we used the same planar hypothesis to search for a similar behaviour
in our sample. Using the results from method 1 (Sect. 2.1), we searched for
the best plane containing the 3D velocity vectors by minimising the average
ratio Vz/Vt, that is, the perpendicular (Vz) to the tangential components (Vt)
of the vectors to a given plane in the space (for details, see Sect. 4.1 and
Fig. 3 in Sanna et al. (2017)). The plane is identified with two angles:
imodel, the angle between the perpendicular to the plane and the line of
sight, and PAmodel, the position angle (north to east) of the sky-projection
of the perpendicular to the plane. In Table 3, we list the best fits for our
targets.

We also calculated the inclination angles derived from the best ellipse fits
(assuming the inclined disc-like structure). The inclination, i, is based on
the ratio of minor to major axes in the same convention as imodel, that is,
the angle between the perpendicular to the plane of the best ellipse fit and
the line of sight, and position angle of the minor axis, PAminor, is based on
the ellipses’s fits as given in Table 2 (PAminor$\perp$PAmajor.) We note the
agreement (within 20-40°) between the inclinations of the best plane from the
above model and the plane of the ellipse for G23.207$-$00.377,
G28.817$+$00.365, G31.047$+$00.356, and G31.581$+$00.077. However, in general,
as a result of this comparison, that is, the model of the 3D proper motion
vectors moving along a plane and the maser cloudlets tracing an inclined ring,
we did not find an agreement. This may indicate that the masers are located in
a ring and their motion is substantially out of the plane of the ring. For
G23.389$+$00.185 and G23.657$-$00.127, the fits are mutually exclusive. In the
case of the most circular configuration observed in G23.657$-$00.127, the
plane shows an almost edge-on inclination, while the distribution of maser
spots suggest the existence of a ring-like morphology seen close to face-on.
That may indicate the weakness of a model that does not fit the positions but
only the 3D velocities. However, extensive searches for G23.657$-$00.127 where
several parameters of the model were changed suggest that there is a
significant (dominant) contribution of the line of sight velocities to the 3D
velocities or that the masers move at a large angle with the plane of the sky.
That is generally consistent with a disc-wind scenario (Bartkiewicz et al.,
2020), where although the flow starts from the disc, the velocities have a
significant component along the disc (jet) axis.

Table 3: Parameters of best fits of the preferential planes of gas motion.

Source | imodel | PAmodel | Vsys | i | PAminor
---|---|---|---|---|---
| (°) | (°) | (km s-1) | (°) | (°)
G23.207$-$00.377 | 51 | 165 | 77.5a | 72 |  $+$42
G23.389$+$00.185 | 6 | 15 | 75.5b | 60 |  $-$38
G23.657$-$00.127 | 81 | 285 | 80.5b | 23 |  $+$70
G28.817$+$00.365 | 76 | 285 | 86.9a | 64 |  $-$37
G31.047$+$00.356 | 101 | 200 | 77.6a | 76 |  $-$27
G31.581$+$00.077 | 81 | 245 | 96.0a | 45 |  $-$3
555Systemic velocities are estimated using NH3 emission from a Wienen et al.
(2012), b Urquhart et al. (2011).

##  5 Conclusions

We report the proper motions of 6.7 GHz methanol maser sources that were
earlier classified as having a ring-like morphology (Bartkiewicz et al.,
2009). Three-epoch observations using the EVN and spanning over eight to ten
years enabled us to detect velocities on the order of a few kilometers per
second at distances from 3.19 to 5.5 kpc. We find that in G23.207$-$00.377,
G23.389$+$00.185, G23.657$-$00.127, G28.817$+$00.365, and G31.047$+$00.356,
the internal motions of maser cloudlets clearly suggest expansion from a
common centre. In G31.581$+$00.077, the results are still marginal, and
possible rotation cannot be excluded. The overall morphology of the maser
emission has remained stable, although the intensities of individual maser
cloudlets varied from epoch to epoch, suggesting internal instabilities. Our
studies also show that the maser spot distribution alone does not allow for a
direct interpretation of the ongoing scenario (e.g. if the rings trace the
discs) and that multi-frequency studies at a complementary angular resolution
are needed.

###### Acknowledgements.

AB, MS, PW, AK, MD acknowledge support from the National Science Centre,
Poland through grant 2021/43/B/ST9/02008. The research leading to these
results has received funding from the European Commission Seventh Framework
Programme (FP/2007-2013) under grant agreement No. 283393 (RadioNet3).

## References

  * Bartkiewicz et al. (2008) Bartkiewicz, A., Brunthaler, A., Szymczak, M., van Langevelde, H. J., & Reid, M. J. 2008, A&A, 490, 787
  * Bartkiewicz et al. (2020) Bartkiewicz, A., Sanna, A., Szymczak, M., et al. 2020, A&A, 637, A15
  * Bartkiewicz et al. (2009) Bartkiewicz, A., Szymczak, M., van Langevelde, H. J., Richards, A. M. S., & Pihlström, Y. M. 2009, A&A, 502, 155
  * Breen et al. (2013) Breen, S. L., Ellingsen, S. P., Contreras, Y., et al. 2013, MNRAS, 435, 524
  * De Buizer et al. (2012) De Buizer, J. M., Bartkiewicz, A., & Szymczak, M. 2012, ApJ, 754, 149
  * Fitzgibbon et al. (1999) Fitzgibbon, A., Pilu, M., & Fisher, R. B. 1999, IEEE Trans. Pattern Anal. Mach. Intell., 21, 476
  * Fujisawa et al. (2014) Fujisawa, K., Sugiyama, K., Motogi, K., et al. 2014, PASJ, 66, 31
  * Goddi et al. (2011) Goddi, C., Moscadelli, L., & Sanna, A. 2011, A&A, 535, L8
  * Hu et al. (2016) Hu, B., Menten, K. M., Wu, Y., et al. 2016, ApJ, 833, 18
  * Keimpema et al. (2015) Keimpema, A., Kettenis, M. M., Pogrebenko, S. V., et al. 2015, Experimental Astronomy, 39, 259
  * Matthews et al. (2010) Matthews, L. D., Greenhill, L. J., Goddi, C., et al. 2010, ApJ, 708, 80
  * Menten (1991) Menten, K. M. 1991, ApJ, 380, L75
  * Minier et al. (2000) Minier, V., Booth, R. S., & Conway, J. E. 2000, A&A, 362, 1093
  * Moscadelli et al. (2000) Moscadelli, L., Cesaroni, R., & Rioja, M. J. 2000, A&A, 360, 663
  * Moscadelli et al. (2005) Moscadelli, L., Cesaroni, R., & Rioja, M. J. 2005, A&A, 438, 889
  * Moscadelli et al. (2011a) Moscadelli, L., Cesaroni, R., Rioja, M. J., Dodson, R., & Reid, M. J. 2011a, A&A, 526, A66
  * Moscadelli et al. (2007) Moscadelli, L., Goddi, C., Cesaroni, R., Beltrán, M. T., & Furuya, R. S. 2007, A&A, 472, 867
  * Moscadelli et al. (2011b) Moscadelli, L., Sanna, A., & Goddi, C. 2011b, A&A, 536, A38
  * Nakamura et al. (2023) Nakamura, M., Motogi, K., Nakamura, H., Yonekura, Y., & Fujisawa, K. 2023, MNRAS, 526, 1000
  * Norris et al. (1993) Norris, R. P., Whiteoak, J. B., Caswell, J. L., Wieringa, M. H., & Gough, R. G. 1993, ApJ, 412, 222
  * Phillips et al. (1998) Phillips, C. J., Norris, R. P., Ellingsen, S. P., & McCulloch, P. M. 1998, MNRAS, 300, 1131
  * Reid et al. (2019) Reid, M. J., Menten, K. M., Brunthaler, A., et al. 2019, ApJ, 885, 131
  * Sanna et al. (2019) Sanna, A., Kölligan, A., Moscadelli, L., et al. 2019, A&A, 623, A77
  * Sanna et al. (2010a) Sanna, A., Moscadelli, L., Cesaroni, R., et al. 2010a, A&A, 517, A71
  * Sanna et al. (2010b) Sanna, A., Moscadelli, L., Cesaroni, R., et al. 2010b, A&A, 517, A78
  * Sanna et al. (2017) Sanna, A., Moscadelli, L., Surcis, G., et al. 2017, A&A, 603, A94
  * Sanna et al. (2015) Sanna, A., Surcis, G., Moscadelli, L., et al. 2015, A&A, 583, L3
  * Sugiyama et al. (2014) Sugiyama, K., Fujisawa, K., Doi, A., et al. 2014, A&A, 562, A82
  * Szymczak et al. (2018) Szymczak, M., Olech, M., Sarniak, R., Wolak, P., & Bartkiewicz, A. 2018, MNRAS, 474, 219
  * Tanabe & Yonekura (2023) Tanabe, Y. & Yonekura, Y. 2023, The Astronomer’s Telegram, 16303, 1
  * Torstensson et al. (2011) Torstensson, K. J. E., van Langevelde, H. J., Vlemmings, W. H. T., & Bourke, S. 2011, A&A, 526, A38
  * Urquhart et al. (2011) Urquhart, J. S., Morgan, L. K., Figura, C. C., et al. 2011, MNRAS, 418, 1689
  * Walsh et al. (1998) Walsh, A. J., Burton, M. G., Hyland, A. R., & Robinson, G. 1998, MNRAS, 301, 640
  * Wienen et al. (2012) Wienen, M., Wyrowski, F., Schuller, F., et al. 2012, A&A, 544, A146

##  Appendix A Additional tables and figures

Figure 7: Methanol maser emission at 6.7 GHz detected using the EVN towards
G23.207$-$00.377. Top: Distribution of maser spots in each epoch. The name
corresponds to the Galactic coordinates of the brightest spots listed in Table
1, the (0,0) locations. The colours of the circles relate to the LSR
velocities as shown on the right bar. The dotted ellipses trace the best fits
to all spots detected at each epoch (Table 2). Bottom: Spectra of individual
6.7 GHz maser cloudlets with Gaussian velocity profiles. Each circle traces
the emission of a single maser spot, while the lines represent the fitted
Gaussian profiles. In cases with a complex profile, we also draw the single
Gaussian profiles, denoted by the dotted lines. The cloudlets used for the
proper motion studies are marked by the black fitting lines, and they are
characterised in Table LABEL:clouds. The green lines trace the remaining
cloudlets with their Gaussian profiles, while crosses correspond to spots from
cloudlets without Gaussian characteristics.

Figure 7: Continued but for the G23.389$+$00.185 target.

Figure 7: Continued but for the G28.817$+$00.365 target.

Figure 7: Continued but for the G31.047$+$00.356 target.

Figure 7: Continued but for the G31.581$+$00.077 target.

Figure 8: Dynamic spectra of the variability of the 6.7 GHz methanol maser
lines towards the targets as monitored using the 32-m Torun dish.

Figure 9: Radial (left) and tangential (right) components of the proper
motions of 6.7 GHz methanol cloudlets in targets (as labelled) calculated
relative to the centre of motions marked by the plus signs.

Figure 9: Continued. Figure 10: Histograms of separations between NE and SW
maser spot pairs in each epoch and the separation increase between epochs E2
and E3 relative to E1 in G23.389$+$00.185. The dashed lines mark the mean
histogram values, while the dotted lines mark the median values. Figure 10:
Continued. Histograms of separations between northern and southern maser spot
pairs in each epoch and the separation increase between epochs E2 and E3
relative to E1 in G23.657$-$00.127. Figure 11: Continued. Histograms of
separations between eastern and western maser spot pairs in each epoch and the
separation increase between epochs E2 and E3 relative to E1 in
G28.817$+$00.365. Figure 12: Continued. Histograms of separations between
eastern and western maser spot pairs in each epoch and the separation increase
between epochs E2 and E3 relative to E1 in G31.047$+$00.356. Figure 13:
Continued. Histograms of separations between eastern and western maser spot
pairs in each epoch and the separation increase between epochs E2 and E3
relative to E1 in G31.581+00.077.

Figure 14: Thermal emission closer to G23.389$+$00.185 (top panel), G23.657$-$00.127 (middle panel), and G31.581$+$00.077 (bottom panel) from the ALMA archive. Data were part of the large programme ALMAGal (ID: 2019.1.00195.L). The methanol emission is presented with the green ellipses. The colourful images correspond to C18O (left) and CH3CN (right) emission for channel maps taken at the line peak. Table 4:  Parameters of 6.7 GHz methanol maser cloudlets used in the proper motion studies. The coordinates are first-epoch data and are relative to averaged proper motions of persistent cloudlets with linear motions; the same as in Figs 1-5. The (0,0) points are listed for each target. Cloudlets that are marked by the superscript $n$ were not used in the proper motion calculations due to their non-linear motions. The full-width at the half-maximum is given for the Gaussian fit of each cloudlet; $-$ denotes a non-Gaussian profile. In the case of G31.047$+$00.356, we list parameters for maser spots (see explanations in Sect. 3.1). Data for G23.657$-$00.127 are presented in more detail in Table A.1 in Bartkiewicz et al. (2020) Vp |  $\Delta$RA |  $\Delta$Dec | $\mu_{\rm RA}$ | $\mu_{\rm Dec}$ | Flux density | FWHM
---|---|---|---|---|---|---
(km s-1) | (mas) | (mas) | (mas yr-1) | (mas yr-1) | E1 | E2 | E3 | E1 | E2 | E3
|  |  |  |  | (Jy beam-1) | (km s-1)
G23.207$-$00.377 | RA=18h34m55$\aas@@fstack{s}$20790, Dec=$-$08o49’11$\aas@@fstack{\prime\prime}$8421 (J2000)
72.53 | -38.824 | +14.095 | -0.04 $\pm$ 0.01 | +0.19 $\pm$ 0.01 | 0.319 | 0.204 | 0.365 | 0.35 | 0.39 | 0.30
73.84 | +30.114 | -32.762 | +0.17 $\pm$ 0.01 | +0.09 $\pm$ 0.02 | 0.446 | 1.195 | 0.319 | 0.28 | 0.29 | 0.32
74.85 | +37.313 | -48.534 | +0.16 $\pm$ 0.01 | +0.04 $\pm$ 0.04 | 0.787 | 0.444 | 0.457 | 0.33 | 0.51 | 0.61
|  |  |  |  |  |  |  | 0.56 | 0.53 |
75.07 | +62.927 | -60.272 | +0.05 $\pm$ 0.03 | -0.10 $\pm$ 0.01 | 0.462 | 0.384 | 0.421 | 0.29 | 0.32 | 0.33
75.42 | -206.951 | -75.988 | -0.17 $\pm$ 0.01 | -0.00 $\pm$ 0.01 | 2.040 | 1.343 | 1.183 | 0.36 | 0.39 | 0.39
75.51 | +119.937 | -91.755 | +0.19 $\pm$ 0.01 | -0.15 $\pm$ 0.01 | 1.356 | 1.049 | 0.807 | 0.35 | 0.54 | 0.42
75.69 | -158.935 | -191.421 | -0.21 $\pm$ 0.04 | -0.27 $\pm$ 0.05 | 0.130 | 0.307 | 0.371 | 0.32 | 0.29 | 0.29
76.30n | +50.295 | -54.298 | -0.13 $\pm$ 0.04 | -0.08 $\pm$ 0.01 | 10.212 | 2.264 | 1.914 | 0.40d | 0.36 | 0.35
76.66n | +52.242 | -43.572 | +0.54 $\pm$ 0.11 | +0.22 $\pm$ 0.03 |  | 3.528 | 3.978 | 0.40d | 0.47 | 0.44
77.14 | +62.746 | -49.179 | +0.06 $\pm$ 0.01 | -0.02 $\pm$ 0.01 | 18.302 | 5.211 | 4.615 | 0.36 | 0.50 | 0.65
77.49 | +92.042 | -46.048 | +0.05 $\pm$ 0.01 | -0.11 $\pm$ 0.01 | 7.326 | 3.939 | 4.621 | 0.45 | 0.55 | 0.37
|  |  |  |  |  |  |  |  |  | 0.36
77.93 | +74.636 | -52.639 | -0.03 $\pm$ 0.03 | -0.07 $\pm$ 0.02 | 6.301 | 2.607 | 3.052 | 0.45 | 0.39 | 0.37
|  |  |  |  |  |  |  |  | 0.69 | 0.80
78.59 | +106.129 | -21.073 | -0.05 $\pm$ 0.01 | -0.15 $\pm$ 0.03 | 0.607 | 0.604 | 0.609 | $-$ | 0.41 | 0.42
78.59 | +116.001 | -17.315 | +0.09 $\pm$ 0.01 | -0.08 $\pm$ 0.03 | 0.949 | 0.370 | 0.317 | 0.54 | 0.40 | 0.49
79.20 | +113.382 | -18.202 | +0.05 $\pm$ 0.01 | -0.06 $\pm$ 0.01 | 2.835 | 1.765 | 1.608 | 0.31 | 0.19 | 0.30
79.38 | -73.128 | +141.111 | -0.06 $\pm$ 0.01 | +0.05 $\pm$ 0.01 | 0.709 | 0.268 | 0.340 | 0.23 | 0.32 |  $>$1
|  |  |  |  |  |  |  |  |  | 0.21
79.86 | -94.615 | +184.008 | -0.08 $\pm$ 0.01 | +0.06 $\pm$ 0.04 | 0.260 | 0.298 | 0.329 | 0.22 | 0.33 | 0.31
79.91 | +4.970 | +20.324 | +0.03 $\pm$ 0.01 | +0.11 $\pm$ 0.03 | 0.546 | 2.282 | 1.963 | 0.31 | 0.31 | 0.33
80.78n | -100.734 | +97.145 | -0.20 $\pm$ 0.05 | +0.18 $\pm$ 0.03 | 12.578 | 4.050 | 2.918 | 0.32 | 0.35 | 0.53
|  |  |  |  |  |  |  | 0.72 | 0.33 |
81.84 | -77.854 | +90.433 | -0.15 $\pm$ 0.04 | +0.20 $\pm$ 0.02 | 13.482 | 4.664 | 5.733 | 0.52 | 0.63 | 0.48
81.83 | -94.597 | +95.504 | +0.23 $\pm$ 0.01 | +0.17 $\pm$ 0.01 | 4.189 | 4.162 | 4.647 | 0.35 | $-$ | 0.24
82.76 | -89.469 | +96.954 | -0.47 $\pm$ 0.04 | +0.10 $\pm$ 0.01 | 2.309 | 9.441 | 14.526 | 0.41 | 0.37 | 0.38
|  |  |  |  |  |  |  |  |  | 0.57
84.87 | +14.185 | +62.763 | +0.17 $\pm$ 0.01 | -0.01 $\pm$ 0.01 | 1.584 | 1.920 | 2.485 | 0.34 | 0.40 | 0.40
|  |  |  |  |  |  |  | 0.42 |  |
G23.389$+$00.185 | RA=18h33m14$\aas@@fstack{s}$32364, Dec=$-$08o23’57$\aas@@fstack{\prime\prime}$5461 (J2000)
72.39 | -62.001 | +40.339 | -0.01 $\pm$ 0.01 | -0.03 $\pm$ 0.01 | 7.076 | 4.943 | 6.594 | 0.46 | 0.32 | 0.30
73.62 | +69.944 | +43.180 | +0.26 $\pm$ 0.06 | +0.65 $\pm$ 0.10 | 9.712 | 1.128 | 2.481 | 0.16 | 0.37 | 0.40
|  |  |  |  |  |  |  | 0.51 |  |
74.00 | +65.994 | +36.741 | +0.02 $\pm$ 0.02 | -0.16 $\pm$ 0.06 | 3.215 | 3.944 | 5.206 | 0.40 | 0.25 | 0.27
|  |  |  |  |  |  |  |  | 0.31 |
74.50 | -26.090 | +42.224 | +0.10 $\pm$ 0.01 | -0.08 $\pm$ 0.03 | 5.628 | 2.979 | 3.739 | 0.30 | 0.40 | 0.39
74.81 | +48.317 | +3.084 | -0.02 $\pm$ 0.01 | +0.00 $\pm$ 0.01 | 20.440 | 12.713 | 14.389 | 0.35 | 0.37 | 0.35
|  |  |  |  |  |  |  | 0.42 | 0.33 | 0.40
75.34 | +16.794 | +73.460 | +0.00 $\pm$ 0.01 | +0.08 $\pm$ 0.01 | 23.521 | 14.063 | 16.308 | 0.24 | 0.28 | 0.28
|  |  |  |  |  |  |  |  |  | 0.37
76.04 | -75.817 | -81.790 | -0.12 $\pm$ 0.04 | -0.05 $\pm$ 0.03 | 4.859 | 5.303 | 7.542 | 0.16 |  |
|  |  |  |  |  |  |  | 0.42 | 0.42 | 0.33
|  |  |  |  |  |  |  | 0.29 | 0.33 | 0.36
76.13 | -59.326 | -71.429 | -0.26 $\pm$ 0.08 | -0.26 $\pm$ 0.01 | 6.771 | 4.557 | 5.545 | 0.36 | 0.51 | 0.45
|  |  |  |  |  |  |  |  | 0.14 |
77.01 | +37.558 | -1.593 | +0.04 $\pm$ 0.01 | -0.03 $\pm$ 0.02 | 0.693 | 0.338 | 0.396 | 0.30 | 0.31 | 0.30
77.45 | -15.369 | -84.216 | -0.01 $\pm$ 0.01 | -0.13 $\pm$ 0.01 | 0.533 | 0.216 | 0.306 | 0.23 | 0.27 | 0.26
G28.817$+$00.365 | RA=18h42m37$\aas@@fstack{s}$34596, Dec=$-$03o29’40$\aas@@fstack{\prime\prime}$9556 (J2000)
87.73 | -42.815 | -22.111 | -0.04 $\pm$ 0.04 | +0.07 $\pm$ 0.05 | 0.306 | 0.178 | 0.211 | $-$ | 0.23 | 0.23
89.75 | +28.677 | +27.623 | -0.06 $\pm$ 0.00 | -0.33 $\pm$ 0.07 | 0.660 | 0.264 | 0.278 | 0.68 | 0.37 | 0.64
|  |  |  |  |  |  |  |  |  | 0.25
90.01 | -60.296 | +14.278 | -0.01 $\pm$ 0.05 | -0.07 $\pm$ 0.02 | 0.124 | 0.085 | 0.089 | $-$ | $-$ | 0.21
90.45n | +29.833 | +33.252 | +0.07 $\pm$ 0.01 | +0.10 $\pm$ 0.01 | 2.774 | 1.245 | 1.479 | 0.56 | 0.54 | 0.31
91.33 | +33.482 | +52.819 | +0.03 $\pm$ 0.01 | +0.31 $\pm$ 0.08 | 2.309 | 1.360 | 1.365 | 0.32 | 0.22 | 0.26
|  |  |  |  |  |  |  |  | 0.31 | 0.26
|  |  |  |  |  |  |  |  | 0.32 | 0.32
92.39n | +24.487 | +43.176 | -0.17 $\pm$ 0.14 | +0.57 $\pm$ 0.18 | 0.371 | 0.209 | 0.194 | 0.43 | 0.88 | 0.27
|  |  |  |  |  |  |  | 0.44 |  | 0.40
92.65 | +27.411 | +60.646 | +0.09 $\pm$ 0.01 | +0.03 $\pm$ 0.03 | 1.924 | 1.129 | 1.292 | 0.38 | 0.40 | 0.42
G31.047$+$00.356 (spots) | RA=18h46m43$\aas@@fstack{s}$85452, Dec=$-$01o30’54$\aas@@fstack{\prime\prime}$1557 (J2000)
78.07 | +7.095 | +8.027 | -0.02 $\pm$ 0.01 | +0.02 $\pm$ 0.05 | 0.255 | 0.108 | 0.068 |  |  |
78.25 | +6.715 | +6.818 | -0.03 $\pm$ 0.01 | -0.01 $\pm$ 0.01 | 0.045 | 0.122 | 0.111 | $-$ | 0.46 | 0.38
78.42n | +7.633 | +5.233 | -0.20 $\pm$ 0.11 | +0.14 $\pm$ 0.07 | 0.122 | 0.067 | 0.114 |  |  |
78.60n | +6.881 | +4.722 | -0.11 $\pm$ 0.06 | +0.10 $\pm$ 0.05 | 0.076 | 0.082 | 0.073 |  |  |
78.78 | +6.638 | +4.113 | -0.01 $\pm$ 0.03 | +0.09 $\pm$ 0.01 | 0.300 | 0.228 | 0.174 |  |  |
78.95 | +6.758 | +3.839 | -0.00 $\pm$ 0.01 | +0.08 $\pm$ 0.01 | 0.610 | 0.540 | 0.492 |  |  |
79.13 | +6.625 | +3.621 | +0.06 $\pm$ 0.02 | +0.06 $\pm$ 0.01 | 0.981 | 0.826 | 0.996 | 0.71 | 0.69 | 0.71
79.30 | +6.593 | +3.584 | +0.07 $\pm$ 0.02 | +0.02 $\pm$ 0.01 | 0.943 | 0.756 | 0.930 |  |  |
79.48 | +6.807 | +3.180 | +0.03 $\pm$ 0.00 | +0.01 $\pm$ 0.02 | 0.641 | 0.562 | 0.689 |  |  |
79.65 | +7.274 | +3.109 | -0.01 $\pm$ 0.01 | -0.08 $\pm$ 0.05 | 0.433 | 0.463 | 0.597 |  |  |
79.83 | +7.893 | +2.172 | +0.00 $\pm$ 0.00 | -0.03 $\pm$ 0.03 | 0.278 | 0.604 | 0.641 |  |  |
80.00 | +8.222 | +2.092 | +0.02 $\pm$ 0.01 | -0.04 $\pm$ 0.02 | 0.382 | 1.090 | 1.040 | 0.34 | 0.60 | 0.54
80.18 | +8.406 | +2.072 | +0.03 $\pm$ 0.01 | -0.03 $\pm$ 0.03 | 0.471 | 1.657 | 1.880 |  |  |
80.36 | +8.650 | +2.197 | -0.00 $\pm$ 0.02 | -0.08 $\pm$ 0.06 | 0.467 | 1.487 | 1.763 |  |  |
80.53 | +8.212 | +1.292 | +0.02 $\pm$ 0.00 | -0.03 $\pm$ 0.02 | 0.943 | 1.453 | 1.464 |  |  |
80.71 | +7.973 | +0.656 | +0.01 $\pm$ 0.01 | -0.04 $\pm$ 0.01 | 1.580 | 1.664 | 1.867 | 0.81 | 0.21 | 0.18
80.88 | +7.830 | -0.395 | +0.02 $\pm$ 0.01 | -0.03 $\pm$ 0.01 | 1.351 | 1.416 | 1.584 |  | 0.65 | 0.61
81.06 | +7.427 | -0.946 | +0.04 $\pm$ 0.01 | -0.05 $\pm$ 0.01 | 1.142 | 1.274 | 1.484 |  |  |
81.06 | -51.775 | -28.434 | +0.17 $\pm$ 0.02 | +0.07 $\pm$ 0.05 | 0.198 | 0.232 | 0.357 | $-$ | 0.29 | 0.29
81.23 | +6.718 | -1.695 | +0.06 $\pm$ 0.01 | -0.05 $\pm$ 0.01 | 0.920 | 0.903 | 0.965 |  |  |
81.41 | +5.790 | -2.762 | +0.12 $\pm$ 0.01 | +0.02 $\pm$ 0.01 | 0.357 | 0.393 | 0.393 |  |  |
82.82 | -39.990 | -6.322 | -0.28 $\pm$ 0.01 | +0.07 $\pm$ 0.01 | 0.115 | 2.263 | 3.496 |  |  |
82.99 | -39.859 | -6.220 | -0.30 $\pm$ 0.01 | +0.04 $\pm$ 0.01 | 0.528 | 1.019 | 0.560 | 0.28 | 0.29 | 0.31
G31.581$+$00.077 | RA=18h48m41.94582, Dec=$-$01o10’2$\aas@@fstack{\prime\prime}$4794 (J2000)
91.60n | +34.405 | -128.074 | -0.00 $\pm$ 0.11 | +0.21 $\pm$ 0.06 | 0.299 | 0.102 | 0.110 | 0.31 | 0.34 | 0.37
95.64 | -71.099 | -48.651 | -0.01 $\pm$ 0.00 | -0.04 $\pm$ 0.00 | 2.630 | 1.083 | 1.822 | 0.33 | 0.34 | 0.33
97.75 | -132.089 | +8.874 | -0.01 $\pm$ 0.00 | +0.06 $\pm$ 0.00 | 0.266 | 0.132 | 0.211 | $-$ | 0.24 | 0.22
98.01n | -33.952 | -55.679 | -0.01 $\pm$ 0.06 | +0.04 $\pm$ 0.02 | 0.741 | 0.202 | 0.237 | 0.29 | 0.93∗ | 0.30
98.45 | -38.817 | -52.326 | +0.07 $\pm$ 0.04 | -0.10 $\pm$ 0.05 | 0.319 | 0.141 | 0.247 | 0.36 | 0.93∗ | 0.31
98.80 | +78.855 | +45.003 | -0.04 $\pm$ 0.01 | +0.02 $\pm$ 0.03 | 1.628 | 1.078 | 2.020 | 0.30 | 0.39 | 0.33
99.24 | +82.712 | +39.239 | -0.04 $\pm$ 0.03 | +0.07 $\pm$ 0.06 | 0.847 | 0.367 | 0.737 | 0.30 | 0.32 | 0.31
99.68 | +80.440 | +7.862 | +0.03 $\pm$ 0.03 | -0.01 $\pm$ 0.02 | 0.749 | 0.517 | 1.081 | 0.37 | 0.45 | 0.40
Table 4: Continued. 666d in G23.207$-$00.377 denotes a case when a single
Gaussian profile in the first epoch (E1) becomes a double in E2 and E3.