# On the structure of weak solutions to the Riemann problem for degenerate
nonlinear diffusion equation

Evgeny Yu. Panov
Yaroslav-the-Wise Novgorod State University, Veliky Novgorod, Russian
Federation,
Research and Development Center, Veliky Novgorod, Russian Federation and
Peoples’ Friendship University of Russia (RUDN University), Moscow, Russian
Federation

###### Abstract

We find an explicit form of weak solutions to a Riemann problem for a
degenerate semilinear parabolic equation with piecewise constant diffusion
coefficient. It is demonstrated that the phase transition lines (free
boundaries) correspond to the minimum point of some strictly convex function
of a finite number of variables. In the limit as number of phases tend to
infinity we obtain a variational formulation of self-similar solution with an
arbitrary nonnegative diffusion function.

In a half-plane $t>0$, $x\in{\mathbb{R}}$, we consider a semilinear parabolic
equation

$u_{t}=(a^{2}(u)u_{x})_{x}=A(u)_{xx},$ (1)

where $A^{\prime}(u)=a^{2}(u)$, $a(u)\in L^{\infty}({\mathbb{R}})$, $a(u)\geq
0$. The change $U=A(u)$ reduces (1) to the equation

$b(U)_{t}=U_{xx},$

with $b(U)=A^{-1}(U)$ being a strictly increasing and possibly discontinuous
function.

The same reduction was used in [2, Chapter 5,§ 9] for a Stefan problem. By the
similar methods as in [2] one can establish the existence and uniqueness of a
bounded weak solution (understood in the sense of distributions) of the Cauchy
problem for equation (1). In the case of piecewise smooth weak solution the
following conditions should be fulfilled on discontinuity lines $x=x(t)$:

$[A(u)]=0,\quad[u]x^{\prime}(t)+[A(u)_{x}]=0,$ (2)

where $[v]=v(t,x(t)+)-v(t,x(t)-)$ denotes a jump of a function $v=v(t,x)$ on
the discontinuity line. Conditions (2) are derived by the application of the
distribution $0=u_{t}-A(u)_{xx}$ to an arbitrary test function and integration
by parts with the help of Green formula. As is easy to verify, these
conditions together with the requirement that $u(t,x)$ is a classic solution
of (1) in the smoothness domains are equivalent to the statement that
$u=u(t,x)$ is a weak solution of equation (1).

We will study the Cauchy problem for equation (1) with a Riemann initial data

$u(0,x)=\left\\{\begin{array}[]{lr}u_{-},&x<0,\\\
u_{+},&x>0.\end{array}\right.$ (3)

As was mentioned above, a weak solution of (1), (3) is unique. By the
uniqueness and the invariance of the problem under the transformation group
$(t,x)\to(\lambda^{2}t,\lambda x)$, $\lambda\in{\mathbb{R}}$, $\lambda\not=0$,
a weak solution of problem (1), (3) is self-similar: $u(t,x)=v(\xi)$,
$\xi=x/\sqrt{t}$. In the case of the heat equation $u_{t}=a^{2}u_{xx}$ a self-
similar solution $u=v(\xi)$ must satisfy the linear ODE
$a^{2}v^{\prime\prime}=-\xi v^{\prime}/2$, the general solution of which is
$v=C_{1}F(\xi/a)+C_{2}$, $C_{1},C_{2}=\mathrm{const}$, where

$F(\xi)=\frac{1}{2\sqrt{\pi}}\int_{-\infty}^{\xi}e^{-s^{2}/4}ds$

is the error function. In particular, the solution of Riemann problem (1), (3)
can be found if we choose $C_{2}=u_{-}$, $C_{1}=u_{+}-u_{-}$. We consider now
the nonlinear case assuming that the diffusion coefficient is a piecewise
constant function, $a(u)=a_{k}$ for $u_{k}<u<u_{k+1}$, $k=0,\ldots,n$, where

$u_{-}=u_{0}<u_{1}<\cdots<u_{n}<u_{n+1}=u_{+}$

(since equation (1) is invariant under the change $x\to-x$, we can suppose
that $u_{+}>u_{-}$), and $a_{k+1}\not=a_{k}$, $k=0,\ldots,n-1$. Let us begin
with the nondegenerate case $a_{k}>0$, $k=0,\ldots,n$. We are going to find a
weak solution $u=v(\xi)$ of (1), (3) as a piecewise smooth function

$\displaystyle
v(\xi)=u_{k}+\frac{u_{k+1}-u_{k}}{F(\xi_{k+1}/a_{k})-F(\xi_{k}/a_{k})}(F(\xi/a_{k})-F(\xi_{k}/a_{k})),$
(4) $\displaystyle\xi_{k}<\xi<\xi_{k+1},\ k=0,\ldots,n,$

where

$-\infty=\xi_{0}<\xi_{1}<\cdots<\xi_{n}<\xi_{n+1}=+\infty$

and we agree that $F(-\infty)=0$, $F(+\infty)=1$. The parabolas $\xi=\xi_{k}$,
$k=1,\ldots,n$, should be determine by conditions (2). Observing that the
piecewise linear function $A(u)$ is strictly increasing, we conclude that the
condition $[A(u)]=0$ reduces to the continuity condition $[u]=0$. We see that
$u=u_{k}$ on the phase transition lines $\xi=\xi_{k}$, and these lines are
weak discontinuities of $u(t,x)$. The second condition in (2) means that
$[A(u)^{\prime}_{\xi}]=0$ (i.e., $A(u)\in C^{1}$). In view of (4) these
conditions reduce to the equalities

$\frac{a_{k}(u_{k+1}-u_{k})F^{\prime}(\xi_{k}/a_{k})}{F(\xi_{k+1}/a_{k})-F(\xi_{k}/a_{k})}=\frac{a_{k-1}(u_{k}-u_{k-1})F^{\prime}(\xi_{k}/a_{k-1})}{F(\xi_{k}/a_{k-1})-F(\xi_{k-1}/a_{k-1})},\quad
k=1,\ldots,n.$ (5)

To prove that the nonlinear system (5) admits a solution, we observe that it
is equivalent to the equality $\nabla E(\bar{\xi})=0$, where the function

$E(\bar{\xi})=-\sum_{k=0}^{n}(a_{k})^{2}(u_{k+1}-u_{k})\ln(F(\xi_{k+1}/a_{k})-F(\xi_{k}/a_{k})),\quad\bar{\xi}=(\xi_{1},\ldots,\xi_{n})\in\Omega,$
(6)

defined in the convex open domain $\Omega$ given by the inequalities
$\xi_{1}<\cdots<\xi_{n}$. Obviously, $E(\bar{\xi})\in C^{\infty}(\Omega)$.
Notice that for all $k=0,\ldots,n$

$\ln(F(\xi_{k+1}/a_{k})-F(\xi_{k}/a_{k}))<0.$ (7)

In particular, $E(\bar{\xi})>0$. We will call the function $E(\bar{\xi})>0$
the entropy, it depends only on discontinuities of the function $v(\xi)$.

######  Lemma 1.

The sets $E(\bar{\xi})\leq c$ are compact for each constant
$c\in{\mathbb{R}}$.

###### Proof.

If $E(\bar{\xi})\leq c$ then it follows from (7) that for all $k=0,\ldots,n$

$\displaystyle-(a_{k})^{2}(u_{k+1}-u_{k})\ln(F(\xi_{k+1}/a_{k})-F(\xi_{k}/a_{k}))\leq
E(\bar{\xi})\leq c,$

which implies the estimate

$F(\xi_{k+1}/a_{k})-F(\xi_{k}/a_{k})\geq\delta\doteq\exp(-c/m)>0,$ (8)

where $\displaystyle m=\min_{k=0,\ldots,n}(a_{k})^{2}(u_{k+1}-u_{k})>0$. In
the particular cases $k=0,n$ these inequalities imply that
$F(\xi_{1}/a_{0})\geq\delta$,
$F(-\xi_{n}/a_{n})=1-F(\xi_{n}/a_{n})\geq\delta$, therefore
$-r\leq\xi_{1}<\xi_{n}\leq r$, where $r$ is a positive constant satisfying the
condition $\max(F(-r/a_{0}),F(-r/a_{n}))\leq\delta$. Since the remaining
coordinates of $\bar{\xi}$ are situated between $\xi_{1}$ and $\xi_{n}$, we
find that ${\|\bar{\xi}\|_{\infty}\leq r}$. Further, since
$F^{\prime}(x)=\frac{1}{2\sqrt{\pi}}e^{-x^{2}/4}<1$, the function $F(x)$ is
Lipschitz with constant $1$ and it follows from (8) that

$(\xi_{k+1}-\xi_{k})/a_{k}\geq
F(\xi_{k+1}/a_{k})-F(\xi_{k}/a_{k})\geq\delta,\quad k=1,\ldots,n-1.$

We find that $\xi_{k+1}-\xi_{k}\geq\delta_{1}=\delta\min a_{k}$. We conclude
tat the set $E(\bar{\xi})\leq c$ lies in the compact set

$K=\\{\ \bar{\xi}=(\xi_{1},\ldots,\xi_{n})\in{\mathbb{R}}^{n}\ |\
\|\bar{\xi}\|_{\infty}\leq r,\ \xi_{k+1}-\xi_{k}\geq\delta_{1}\ \forall
k=1,\ldots,n-1\ \\}.$

By the continuity of $E(\bar{\xi})$ the set $E(\bar{\xi})\leq c$ is a closed
subset of $K$ and therefore is compact. ∎

We take $c>N\doteq\inf E(\bar{\xi})$. Then the set $E(\bar{\xi})\leq c$ is not
empty. By Lemma 1 this set is compact and therefore the continuous function
$E(\bar{\xi})$ reaches the minimal value on it, which is evidently equal to
$N$. We proved the existence of global minimum $E(\bar{\xi}_{0})=\min
E(\bar{\xi})$. At the minimum point $\bar{\xi}_{0}$ the required condition
$\nabla E(\bar{\xi}_{0})=0$ holds and we conclude that system (5) has a
solution.

The uniqueness of this solution follows from the uniqueness of a weak solution
of our Riemann problem. Alternatively, this uniqueness can be derived from
strict convexity of the entropy function $E(\bar{\xi})$ (so that this function
can have at most one extremum in $\Omega$). In view of representation (6) the
strict convexity of the entropy easily follows from the lemma below.

######  Lemma 2.

The function $P(x,y)=-\ln(F(x)-F(y))$ is strictly convex in the half-plane
$x>y$.

###### Proof.

The function $P(x,y)$ is infinitely differentiable in the domain $x>y$. To
prove the lemma, we need to establish that the Hessian $D^{2}P$ is positive
definite at every point. By the direct computation we find

$\displaystyle\frac{\partial^{2}}{\partial
x^{2}}P(x,y)=\frac{(F^{\prime}(x))^{2}-F^{\prime\prime}(x)(F(x)-F(y))}{(F(x)-F(y))^{2}},$
$\displaystyle\frac{\partial^{2}}{\partial
y^{2}}P(x,y)=\frac{(F^{\prime}(y))^{2}-F^{\prime\prime}(y)(F(y)-F(x))}{(F(x)-F(y))^{2}},\
\frac{\partial^{2}}{\partial x\partial
y}P(x,y)=-\frac{F^{\prime}(x)F^{\prime}(y)}{(F(x)-F(y))^{2}}.$

We have to prove positive definiteness of the matrix
$Q=(F(x)-F(y))^{2}D^{2}P(x,y)$ with the components

$\displaystyle Q_{11}=(F^{\prime}(x))^{2}-F^{\prime\prime}(x)(F(x)-F(y)),$
$\displaystyle Q_{22}=(F^{\prime}(y))^{2}-F^{\prime\prime}(y)(F(y)-F(x)),\
Q_{12}=Q_{21}=-F^{\prime}(x)F^{\prime}(y).$

Since $F^{\prime}(x)=e^{-x^{2}/4}$, then
$F^{\prime\prime}(x)=-\frac{x}{2}F^{\prime}(x)$ and the diagonal elements of
this matrix can be written in the form

$\displaystyle Q_{11}=F^{\prime}(x)(\frac{x}{2}(F(x)-F(y))+F^{\prime}(x))=$
$\displaystyle
F^{\prime}(x)(\frac{x}{2}(F(x)-F(y))+(F^{\prime}(x)-F^{\prime}(y)))+F^{\prime}(x)F^{\prime}(y),$
$\displaystyle
Q_{22}=F^{\prime}(y)(\frac{y}{2}(F(y)-F(x))+(F^{\prime}(y)-F^{\prime}(x)))+F^{\prime}(x)F^{\prime}(y).$

By Cauchy mean value theorem there exists such a value $z\in(y,x)$ that

$\frac{F^{\prime}(x)-F^{\prime}(y)}{F(x)-F(y)}=\frac{F^{\prime\prime}(z)}{F^{\prime}(z)}=-z/2.$

Therefore,

$\displaystyle
Q_{11}=F^{\prime}(x)(F(x)-F(y))(x-z)/2+F^{\prime}(x)F^{\prime}(y),$
$\displaystyle
Q_{22}=F^{\prime}(y)(F(x)-F(y))(z-y)/2+F^{\prime}(x)F^{\prime}(y),$

and it follows that $Q=R_{1}+F^{\prime}(x)F^{\prime}(y)R_{2}$, where $R_{1}$
is a diagonal matrix with the positive diagonal elements
$F^{\prime}(x)(F(x)-F(y))(x-z)/2$, $F^{\prime}(y)(F(x)-F(y))(z-y)/2$ while
$R_{2}=\left(\begin{smallmatrix}1&-1\\\ -1&1\end{smallmatrix}\right)$. Since
$R_{1}>0$, $R_{2}\geq 0$, then the matrix $Q>0$, as was to be proved. ∎

Now we consider the case when some of the coefficients $a_{k}=0$. If $k=0$ or
$k=n$, the structure (4) of the solution remains the same but for
$\xi<\xi_{1}$ (respectively, for $\xi>\xi_{n}$) the solution becomes constant:
$v\equiv u_{-}$ ($v\equiv u_{+}$). Moreover, the discontinuity $\xi=\xi_{1}$
($\xi=\xi_{n}$) is now strong, and condition (2) implies the following
relation of Stefan type

$-(u_{1}-u_{-})\xi_{1}/2=\frac{a_{1}(u_{2}-u_{1})F^{\prime}(\xi_{1}/a_{1})}{F(\xi_{2}/a_{1})-F(\xi_{1}/a_{1})},$

respectively,

$(u_{+}-u_{n})\xi_{n}/2=\frac{a_{n-1}(u_{n}-u_{n-1})F^{\prime}(\xi_{n}/a_{n-1})}{F(\xi_{n}/a_{n-1})-F(\xi_{n-1}/a_{n-1})}.$

These relations replace, respectively, the first and the last equality in (5).
Notice that the first condition $[A(u)]=0$ in (2) is satisfied because the
function $A(u)$ is constant on the segment $[u_{-},u_{1}]$ (on
$[u_{n},u_{+}]$). Replacing the first term in sum (6) by
$(u_{1}-u_{-})(\xi_{1})^{2}/4$ (the last term by
$(u_{+}-u_{n})(\xi_{n})^{2}/4$), we obtain that all the required conditions on
the lines $\xi=\xi_{k}$, $k=1,\ldots,n$, reduce again to the condition $\nabla
E=0$, and the solution is uniquely determined by the minimum point of the
entropy $E(\bar{\xi})$. Remark that after the described reduction the entropy
function remains to satisfy the statements of Lemma 1, Lemma 2.

The case when the diffusion degenerates in an inner interval, i.e., when
$a_{k}=0$ for $0<k<n$, is more complicated. In this case two weak
discontinuities $\xi=\xi_{k},\xi_{k+1}$ merge into one strong discontinuity
$\xi=\xi_{k}$ with the limit values $v(\xi_{k}-)=u_{k}$,
$v(\xi_{k}+)=u_{k+1}$, (one just need to put $\xi_{k+1}=\xi_{k}$ in formula
(4)). Condition (2) on the line $\xi=\xi_{k}$ reduces to the equality

$-(u_{k+1}-u_{k})\xi_{k}/2=\frac{a_{k+1}(u_{k+2}-u_{k+1})F^{\prime}(\xi_{k}/a_{k+1})}{F(\xi_{k+2}/a_{k+1})-F(\xi_{k}/a_{k+1})}-\frac{a_{k-1}(u_{k}-u_{k-1})F^{\prime}(\xi_{k}/a_{k-1})}{F(\xi_{k}/a_{k-1})-F(\xi_{k-1}/a_{k-1})}.$

This condition replaces two conditions with numbers $k,k+1$ in system (5),
where we also agree that $\xi_{k+1}=\xi_{k}$. Generally, we get the system of
$n-l$ equations with $n-l$ unknowns, where $l$ is a number of inner intervals
$(u_{k},u_{k+1})$ with degenerate diffusion ($a_{k}=0$). As above, we find
that solutions of this system coincide with critical points of the entropy
function $E(\bar{\xi})$ of $n-l$ variables defined by the expression

$\displaystyle
E(\xi_{1},\ldots,\xi_{n})=-\sum_{k=0,\ldots,n,a_{k}>0}(a_{k})^{2}(u_{k+1}-u_{k})\ln(F(\xi_{k+1}/a_{k})-F(\xi_{k}/a_{k}))$
$\displaystyle+\sum_{k=0,\ldots,n,a_{k}=0}(u_{k+1}-u_{k})(\xi_{k})^{2}/4,$ (9)

where variables $\xi_{k}$, $\xi_{k+1}$ are identified whenever $a_{k}=0$,
$0<k<n$.

In the same way as in the nondegenerate case we can easily prove that this
function is strictly convex and reaches the unique minimal value
$E(\bar{\xi}_{0})$. Coordinates of the minimum point $\bar{\xi}_{0}$ determine
the unique weak solution of problem (1), (3).

######  Remark 1.

Adding to the entropy (On the structure of weak solutions to the Riemann
problem for degenerate nonlinear diffusion equation) the constant

$\sum_{k=0,\ldots,n,a_{k}>0}(a_{k})^{2}(u_{k+1}-u_{k})\ln((u_{k+1}-u_{k})/a_{k}),$

we obtain the alternative variant of the entropy

$\displaystyle
E_{1}(\bar{\xi})=-\sum_{k=0,\ldots,n,a_{k}>0}(a_{k})^{2}(u_{k+1}-u_{k})\ln\left(\frac{F(\xi_{k+1}/a_{k})-F(\xi_{k}/a_{k})}{(u_{k+1}-u_{k})/a_{k}}\right)$
$\displaystyle+\sum_{k=0,\ldots,n,a_{k}=0}(u_{k+1}-u_{k})(\xi_{k})^{2}/4.$
(10)

If we consider the values $a_{k}$ as a piecewise constant approximation of an
arbitrary diffusion function $a(u)\geq 0$ then, passing in (1) to the limit as
$\max(u_{k+1}-u_{k})\to 0$, we find that the entropy $E_{1}(\bar{\xi})$ turns
into the variational functional

$J(\xi)=-\int_{\\{u\in[u_{-},u_{+}],a(u)>0\\}}(a(u))^{2}\ln(F^{\prime}(\xi(u)/a(u))\xi^{\prime}(u))du+\int_{\\{u\in[u_{-},u_{+}],a(u)=0\\}}(\xi(u))^{2}/4du,$

where $\xi(u)$ is an increasing function on $[u_{-},u_{+}]$, which is expected
to be the inverse function to a self-similar solution $u=u(\xi)$ to problem
(1), (3). Taking into account that

$\ln(F^{\prime}(\xi(u)/a(u))\xi^{\prime}(u))=\ln
F^{\prime}(\xi(u)/a(u))+\ln\xi^{\prime}(u)=-\frac{(\xi(u))^{2}}{4a^{2}(u)}+\ln\xi^{\prime}(u),$

we may simplify the expression for the functional $J(\xi)$

$J(\xi)=-\int_{u_{-}}^{u_{+}}(a(u))^{2}\ln(\xi^{\prime}(u))du+\frac{1}{4}\int_{u_{-}}^{u_{+}}(\xi(u))^{2}du.$
(11)

We see that this functional is strictly convex. As is easy to verify, the
corresponding Euler-Lagrange equation has the form

$\xi(u)/2+((a(u))^{2}/\xi^{\prime}(u))^{\prime}=0.$ (12)

Since $u^{\prime}(\xi)=1/\xi^{\prime}(u)$, $u=u(\xi)$, we can transform (12)
as follows

$\xi(u)/2+((a(u))^{2}u^{\prime}(\xi))^{\prime}_{u}=0.$

Multiplying this equation by $u^{\prime}(\xi)$, we obtain the equation

$(a^{2}u^{\prime})^{\prime}=-\xi u^{\prime}/2,\quad u=u(\xi),$

which is exactly our equation (1) written in the self-similar variable. We
conclude that minimization of the functional (11) gives a variational
formulation of self-similar solutions of equation (1).

## Acknowledgments

The author thanks professor E. V. Radkevich for fruitful discussions. The
research was supported by the Russian Science Foundation, grant 22-21-00344.

## References

  * [1]
  * [2] O.A. Ladyzhenskaya, V.A. Solonnikov and N.N. Ural’tseva, Linear and Quasi-Linear Equations of Parabolic Type, AMS, Providence, RI, 1968.


11institutetext: Zhejiang University of Technology, Hangzhou, 310023, P.R.
China 11email<EMAIL_ADDRESS>
22institutetext: Ping An Life Insurance Of China, Ltd, Shanghai, 200120, P.R.
China 22email<EMAIL_ADDRESS>
33institutetext: Zhejiang University, Hangzhou, 310027, P.R. China
33email<EMAIL_ADDRESS>
# Mid-level Representation Enhancement and Graph Embedded Uncertainty
Suppressing for Facial Expression Recognition

Jie Lei 11 Zhao Liu 22 Zeyu Zou 22 Tong Li 22 Xu Juan 22 Shuaiwei Wang 11
Guoyu Yang 11 Zunlei Feng 33

###### Abstract

Facial expression is an essential factor in conveying human emotional states
and intentions. Although remarkable advancement has been made in facial
expression recognition (FER) task, challenges due to large variations of
expression patterns and unavoidable data uncertainties still remain. In this
paper, we propose mid-level representation enhancement (MRE) and graph
embedded uncertainty suppressing (GUS) addressing these issues. On one hand,
MRE is introduced to avoid expression representation learning being dominated
by a limited number of highly discriminative patterns. On the other hand, GUS
is introduced to suppress the feature ambiguity in the representation space.
The proposed method not only has stronger generalization capability to handle
different variations of expression patterns but also more robustness to
capture expression representations. Experimental evaluation on Aff-Wild2 have
verified the effectiveness of the proposed method. We achieved 2nd place in
the Learning from Synthetic Data (LSD) Challenge of the 4th Competition on
Affective Behavior Analysis in-the-wild (ABAW). The code will be released at
https://github.com/CruiseYuGH/GUS.

##  1 Introduction

Facial expression recognition (FER) is a highly focused field in computer
vision for its potential use in sociable robots, medical treatment, driver
fatigue surveillance, and many other human-computer interaction systems.
Typically, the FER task is to classify an input facial image into one of the
following six basic categories: _anger_ (AN), _disgust_ (DI), _fear_ (FE),
_happiness_ (HA), _sadness_ (SA), and _surprise_ (SU).

In the field of facial expression recognition (FER), various FER systems have
been explored to encode expression information from facial representations.
Several studies have proposed well-designed auxiliary modules to enhance the
foundation architecture of deep models [22, 17, 2, 1, 26]. Yao _et al._ [22]
proposed HoloNet with three critical considerations in the network design.
Zhou _et al._ [26] introduced an spatial-temporal facial graph to encode the
information of facial expressions. Another area focuses on facial expression
data for robust recognition [19, 24, 25, 21, 5]. In [24], the authors proposed
an end-to-end trainable LTNet to discover the latent truths with the
additional annotations from different datasets. Zhao _et al._ [25] presented a
novel peak-piloted deep network (PPDN) that used the peak expression (easy
sample) to supervise the non-peak expression (hard sample) of the same type
and from the same subject.

Despite the promising results, prior approaches tend to learn a limited number
of highly discriminative patterns and suffer from unavoidable expression data
uncertainties. To address these issues, we propose to mix the mid-level
representation with samples from other expressions in the learning process,
thus suppressing the over-activation to some partial expression patterns and
enriching the pattern combination to enhance classifier generalization. In
addition, we introduce graph embedded uncertainty suppressing to reduce the
feature ambiguity in the representation space.

The competition on Affective Behavior Analysis in-the-wild (ABAW) has been
successfully held three times, in conjunction with IEEE FG 2020, ICCV 2021,
and CVPR 2022. The goal of our work is to study facial expression recognition
based on Aff-Wild2 [8, 12, 16, 10, 15, 14, 13, 9, 23, 11, 7] in the 4th
competition. We adopted the proposed mid-level representation enhancement
(MRE) and graph embedded uncertainty suppressing (GUS) for this FER task.
Experiments on Aff-Wild2 show the effectiveness of the proposed method.

##  2 Method

###  2.1 Overview

The overall architecture of our proposed method is illustrated in Fig. 1. It
mainly consists of two parts, _e.g._ , mid-level representation enhancement
and graph embedded uncertainty suppressing. In order to further enhance the
robustness of our framework, we adopt the model ensemble strategy by combining
DMUE [4] with MRE and GUS.

Figure 1: Framework of the proposed MRE and GUS.

###  2.2 Mid-level Representation Enhancement

The mid-level representation can support robust high-level representation to
improve accuracy [3]. However, the learned mid-level representation tends to
be dominated by a limited number of highly discriminative partial patterns,
making false predictions when two expressions share significant similarities
in specific facial parts that are regular in one category.

For facial image $I_{i}$, based on the hierarchical structure of deep neural
networks, we can take the expression representations of mid-level and high-
level according to the layer depth, denoted as $R^{m}_{i}$ and $R^{h}_{i}$,
respectively.

We mix the mid-level representation with a sample from other expressions to
form an new variant. The main idea is to inject partial features as noise data
to improve the generalization capability. Given a facial image $I_{i}$ and a
randomly selected sample $I_{j}$ with a different expression, the mixing
operation can be denoted as:

$\tilde{R_{i}^{m}}(p)=\mathbbm{1}[p\notin\mathcal{N}]R_{i}^{m}(p)+\mathbbm{1}[p\in\mathcal{N}]R_{j}^{m}(p),$
(1)

where $\mathbbm{1}(\cdot)$ is an indicator, $\mathcal{N}$ is the uniform
sampled positions for where the noises are injected, $\tilde{R_{i}^{m}}(p)$
indicates the value of the variant of $R_{i}^{m}$ at position $p$. This
strategy provides a way to allow gradients to suppress the overconfident
partial patterns.

Following the procedure, the variant $\tilde{R_{i}^{m}}$ is obtained, which
helps prevent the network from over-focusing on a small number of
discriminative partial patterns. To amplify this property, we introduce an
expression classification module based on $\tilde{R_{i}^{m}}$ as an additional
branch via global average pooling (GAP) in the network. In this way, the mid-
level expression representation can be enhanced by directly supervised with
the expression label. Thus, the total loss for the FER task is defined as:

$\mathcal{L}_{MRE}=\sum_{i}(\ell(R^{h}_{i},y_{i})+\lambda\ell(\tilde{R_{i}^{m}},y_{i})),$
(2)

where $y_{i}$ is the expression label for $I$, the first term denotes the
original training loss for expression classification towards the high-level
expression representation, the second term denotes the expression
classification loss using the variant mid-level representation on the branch,
and $\lambda$ is a balanced parameter.

###  2.3 Graph Embedded Uncertainty Suppressing

For uncertainty suppressing, we use Graph Convolutional Networks (GCN) [6, 20]
to estabilish semantic relations between the repesentations and reduce the
covariate shifting. The inputs are representation maps
$\hat{F}=\\{\hat{f_{1}},...,\hat{f_{N}}\\}$ extracted by backbone feature
extractor from the original images. To begin with, we use cosine similarity
coefficient to calculated the similarity between different representations as:

$cossim(f_{i},f_{j})=\frac{f_{i}*f_{j}}{||f_{i}||f_{j}||},$ (3)

then, we employ GCN as:

$F^{l+1}=\widetilde{D}^{-\frac{1}{2}}\widetilde{A}\widetilde{D}^{-\frac{1}{2}}F^{l}W^{l},$
(4)

where $\widetilde{A}=A+I$ is the sum of undirected graph A obtained by Eqn.
(3) and the identity matrix, $\widetilde{D}$ is the diagonal matrix from A,
which is $\widetilde{D}_{i,i}=\sum\limits_{j}A_{i,j}$. $F^{l}$ and $F^{l+1}$
are the corresponding input and output representations on the $l_{th}$ level,
and $W^{l}$ are the trainable parameters on this level.

###  2.4 Mixed Loss Functions

In the training process, we have conducted multiple combinations of different
loss functions, finally we use the combination of cross entropy loss $L_{ce}$,
focal loss $L_{fl}$ [18] and sparse regularization loss $L_{fl}$ [27]. Without
loss of generality, the predicted outputs of our model is
$\hat{Y}=\\{\hat{y_{1}},...,\hat{y_{6}}\\}$ the ground truth label is
$Y=\\{y_{1},...y_{6}\\}$, our final loss function can be formulated as:

$L_{fer}=\omega_{1}L_{ce}+\omega_{2}L_{fl}+\omega_{3}L_{sr},$ (5)

where $\omega_{1}$, $\omega_{2}$ and $\omega_{3}$ are balanced parameters.

###  2.5 Predictions Merging and Correcting

We employ a predictions merging strategy to further enhance the robustness of
the whole framework. For the features extracted from different sub-networks,
we use softmax function to calculate the confidential scores of each
expression category, then the confidential scores of each class are summed up
as:

$R_{final}=\omega^{r}_{g}R_{GUS}+\omega^{r}_{m}R_{MRE}+\omega^{r}_{d}R_{DMUE},$
(6)

where weights $\omega^{r}_{g}$, $\omega^{r}_{m}$ and $\omega^{r}_{d}$ are used
to balance the imfluence of different sub-networks in the final predictions.
During the training process we found GUS achieved better results than DMUE and
MRE in all the expression categories except _fear_. As a consequence, we
design two different weights settings, denoted as S1 and S2. S1:
$\omega^{r}_{g}=0.6$, $\omega^{r}_{m}=0.2$ and $\omega^{r}_{d}=0.2$, convicing
more on the predictions of GUS. S2: $\omega^{r}_{g}=0.4$, $\omega^{r}_{m}=0.3$
and $\omega^{r}_{d}=0.3$, relying more on the predictions of MRE, DMUE.
Finally the _argmax_ function is employed to predict the final outputs.

In addition, we design a predictions correcting strategy for the test set. We
adopt a Imagenet pre-trained VGG-19 model to extract features from the test
images, and calculate the cosine similarity between different features. The
images with similarity higher than 0.93 are classified to the same subsets.
During the inference stage, a voting process is carried out in each subset
with more than 2 images: If more than two-thirds of the images are predicted
as the same category, the predicted results of the remaining images are
updated with this category.

##  3 Experiments

###  3.1 Dataset

The Learning from Synthetic Data (LSD) Challenge is part of the 4th Workshop
and Competition on Affective Behavior Analysis in-the-wild (ABAW). For this
Challenge, some specific frames-images from the Aff-Wild2 database have been
selected for expression manipulation. In total, it contains 277,251 synthetic
images annotations in terms of the six basic facial expressions. In addition,
4,670 and 106,121 images from Aff-Wild2 are selected as the validation and
test sets. The synthetic training data have been generated from subjects of
the validation set, but not of the test set.

###  3.2 Settings

We only use the officially provided synthetic images to train the models and
the validation images for finetuning. All the images are resized to 224
$\times$ 224 in these processes. We use random crop, Gaussian blur and random
flip as the data augmentation strategies. We also adopt over-sampling strategy
over the classes with less images to reduce data imbalance. For training the
MRE branch, we use ResNet-50 as the backbone network. As for the DMUE [4] and
GUS branches, we adopt ResNet-18 as the backbone network. All the backbones
are pre-trained on ImageNet. We use SGD with a learning rate of 0.001 during
the optimization in MRE and DMUE, and a learning rate of 0.0001 in GUS.

###  3.3 Results

Table 1 shows the results of the proposed method on the official validation
dataset. By employing DMUE [4], MRE and GUS, we achieve 0.28, 0.14 and 0.32
increase on mean F1-score compared with the baseline model (Resnet-50),
respectively. By applying the model ensemble strategy, we can achieve another
0.03 increase. Meanwhile, the predictions correcting stratagy contributes
another 0.02 increase. Finally, our proposed framework scored 0.8482 on the
validation dataset.

Table 2 shows the results on the test set of 4th ABAW Learning From Synthetic
Data (LSD) Challenge. The five submissions are obtained as follows: (1) Single
GUS module; (2) Predictions merging with ensemble weights S1; (3) Predictions
merging with ensemble weights S2; (4) Predictions merging with ensemble
weights S1 and predictions correcting; (5) Predictions merging with ensemble
weights S2 and predictions correcting. As shown in the table, the 4th
submission owns the highest performance of 36.51 in F1-score, and we achieve
2nd place in the final challenge leaderboard.

Method Class | ANGER | DISGUSS | FEAR | HAPPINESS | SADNESS | SURPRISE | MEAN
---|---|---|---|---|---|---|---
Baseline | 0.4835 | 0.5112 | 0.3860 | 0.5022 | 0.4600 | 0.5731 | 0.4865
DMUE [4] | 0.8235 | 0.8000 | 0.6768 | 0.7438 | 0.7541 | 0.7929 | 0.7642
MRE | 0.7797 | 0.4737 | 0.4521 | 0.7061 | 0.6768 | 0.6809 | 0.6282
GUS | 0.8846 | 0.8814 | 0.5578 | 0.8616 | 0.8399 | 0.8315 | 0.8095
Ensembled | 0.8846 | 0.8814 | 0.6909 | 0.8616 | 0.8399 | 0.8558 | 0.8357
Corrected | 0.8846 | 0.8966 | 0.7143 | 0.8584 | 0.8674 | 0.8676 | 0.8482
Table 1: Performances of different models and strategies on the official validation dataset, where all the results are reported in F1-Score. Submission | Performance Metric
---|---
1 | 31.54
2 | 36.13
3 | 36.33
4 | 36.51
5 | 36.35
Table 2: Performance of 5 submissions on the test set of 4th ABAW Learning
From Synthetic Data Challenge (LSD).

##  4 Conclusion

In this paper, we have proposed the mid-level representation enhancement (MRE)
and graph embedded uncertainty suppressing (GUS) for facial expression
recognition task, aiming at addressing the problem of large variations of
expression patterns and unavoidable data uncertainties. Experiments on Aff-
Wild2 have verified the effectiveness of the proposed method.

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# Discovery of an accretion streamer and a slow wide-angle outflow around FU
Orionis

A.S. Hales National Radio Astronomy Observatory, 520 Edgemont Road,
Charlottesville, VA 22903-2475, United States of America  Joint ALMA
Observatory, Avenida Alonso de Córdova 3107, Vitacura 7630355, Santiago, Chile
Millennium Nucleus on Young Exoplanets and their Moons (YEMS), Chile  A. Gupta
European Southern Observatory, Karl-Schwarzschild-Str. 2, 85748 Garching bei
München, Germany  D. Ruíz-Rodríguez Joint ALMA Observatory, Avenida Alonso de
Córdova 3107, Vitacura 7630355, Santiago, Chile  J. P. Williams Institute for
Astronomy, University of Hawaii, Honolulu, HI 96816, USA  S. Pérez
Departamento de Física, Universidad de Santiago de Chile, Av. Victor Jara
3659, Santiago, Chile  Millennium Nucleus on Young Exoplanets and their Moons
(YEMS), Chile  Center for Interdisciplinary Research in Astrophysics and Space
Exploration (CIRAS), Universidad de Santiago de Chile, Chile  L. Cieza
Instituto de Estudios Astrofísicos, Facultad de Ingeniería y Ciencias,
Universidad Diego Portales, Av. Ejercito 441, Santiago, Chile  Millennium
Nucleus on Young Exoplanets and their Moons (YEMS), Chile  C. González-Ruilova
Instituto de Estudios Astrofísicos, Facultad de Ingeniería y Ciencias,
Universidad Diego Portales, Av. Ejercito 441, Santiago, Chile  Millennium
Nucleus on Young Exoplanets and their Moons (YEMS), Chile  J. E. Pineda Max-
Planck-Institut für extraterrestrische Physik, Giessenbachstrasse 1, D-85748
Garching, Germany  A. Santamaría-Miranda European Southern Observatory, Alonso
de Córdova 3107, Casilla 19, Vitacura, Santiago, Chile  Departamento de
Astronomía, Universidad de Chile, Camino El Observatorio 1515, Las Condes,
Santiago, Chile  J. Tobin National Radio Astronomy Observatory, 520 Edgemont
Road, Charlottesville, VA 22903-2475, United States of America  P. Weber
Departamento de Física, Universidad de Santiago de Chile, Av. Victor Jara
3659, Santiago, Chile  Millennium Nucleus on Young Exoplanets and their Moons
(YEMS), Chile  Center for Interdisciplinary Research in Astrophysics and Space
Exploration (CIRAS), Universidad de Santiago de Chile, Chile  Z. Zhu
Department of Physics and Astronomy, University of Nevada, Las Vegas, 4505
South Maryland Parkway, Las Vegas, NV 89154-4002, USA  Nevada Center for
Astrophysics (NCfA), University of Nevada, Las Vegas, NV, USA  A. Zurlo
Instituto de Estudios Astrofísicos, Facultad de Ingeniería y Ciencias,
Universidad Diego Portales, Av. Ejercito 441, Santiago, Chile  Millennium
Nucleus on Young Exoplanets and their Moons (YEMS), Chile

(Accepted by The Astrophysical Journal)

###### Abstract

We present ALMA 12-m, 7-m & Total Power (TP) Array observations of the FU
Orionis outbursting system, covering spatial scales ranging from 160 to 25,000
au. The high-resolution interferometric data reveals an elongated 12CO(2–1)
feature previously observed at lower resolution in 12CO(3–2). Kinematic
modeling indicates that this feature can be interpreted as an accretion
streamer feeding the binary system. The mass infall rate provided by the
streamer is significantly lower than the typical stellar accretion rates (even
in quiescent states), suggesting that this streamer alone is not massive
enough to sustain the enhanced accretion rates characteristic of the
outbursting class prototype. The observed streamer may not be directly linked
to the current outburst but rather a remnant of a previous, more massive
streamer that may have contributed enough to the disk mass to render it
unstable and trigger the FU Ori outburst. The new data detects, for the first
time, a vast, slow-moving carbon monoxide molecular outflow emerging from this
object. To accurately assess the outflow properties (mass, momentum, kinetic
energy), we employed 13CO(2–1) data to correct for optical depth effects. The
analysis indicates that the outflow corresponds to swept-up material not
associated with the current outburst, similar to slow-molecular outflows
observed around other FUor and Class I protostellar objects.

Star formation; Young stellar objects; FU Orionis stars

††software: Common Astronomy Software Applications (McMullin et al., 2007;
CASA Team et al., 2022), TIPSY (Gupta et al., 2024), and TP2VIS (Koda et al.,
2019)

##  1 Introduction

The flaring of FU Orionis in 1936 (Hoffleit, 1939) marked the discovery of a
new phase in low-mass star formation. FU Ori increased its brightness by over
5 magnitudes within a year (Wachmann, 1954). Herbig (1966) noted that the
brightening could not be explained in terms of a nova event or by sudden
changes in extinction, and that it was most likely due to ‘some phenomenon of
early stellar evolution.’ Hartmann & Kenyon (1985) argued that the increased
luminosity could be explained by a rapid onset of accretion from a rotating
accretion disk (i.e. an accretion outburst). In the accretion disk model, the
energy released by the accreting gas heats up the disk up to several thousand
K, reproducing the increase in luminosity, observed spectral energy
distribution (SED), and peculiar spectral properties such as the apparent
wavelength-dependent stellar spectral type which can be explained by
absorption from different regions of superheated disk atmosphere (Hartmann &
Kenyon, 1996; Hartmann et al., 2016).

For decades FU Ori was the only known star of its class, but the progressive
discovery of more outbursting sources suggested that these are not isolated
cases, but a common, yet short-lived, phase in star formation. Newly
discovered outbursting stars showed diverse outburst properties forcing the
definition of sub-classes: FUors (named after FU Ori) have high-amplitude ($>$
5 mag), long-lived (years to decades) optical bursts (Herbig, 1966), whereas
EXors (misnamed after EX Lupi) have low amplitude (2–4 mag) and shorter (days
to months) periods of enhanced accretion (e.g. Audard et al., 2014, for a
review). The advent of systematic photometric surveys increased the discovery
rate of new eruptive objects (e.g. Kun et al., 2014). Milky Way surveys
indicate that 3-6 $\%$ of Class I protostars show eruptive behavior in a
4-year span and that eruptive events are more common in Class I objects than
in Class II (Contreras Peña et al., 2017).

As new sources are discovered, the heterogeneous nature of their
characteristics presents a significant challenge in terms of classification
and understanding. Fischer et al. (2022) discusses the difficulties
encountered in distinguishing between various types of outbursts, such as FU
Orionis-like events and EX Lupi-like events. These challenges arise due to the
overlapping characteristics exhibited by these phenomena and the limitations
imposed by the available observational data. Recent advancements in time-
dependent, multiwavelength, and spectroscopic analyses suggest a continuum of
diverse phenomena rather than distinct and separable classes. This emphasizes
the need for comprehensive and multifaceted approaches to understand the
complexity of these outburst events fully.

Accretion outbursts are now believed to be central to the formation of low-
mass stars. The well-known Luminosity Problem (Kenyon et al., 1990) arises
from the fact that the measured luminosities of low-mass protostars are
smaller than expected by the quiescent (non-outbursting) mean accretion rates
(10-6 M${}_{\odot}\,$yr-1). The problem is mitigated if the accretion rates
are variable, with stars accreting an important fraction of their mass during
several bursts of enhanced accretion during the Class I phase (Evans et al.,
2009; Dunham et al., 2010) while spending most of their time in lower
quiescent accretion rates than predicted (10-8 M${}_{\odot}\,$yr-1).
Submillimeter monitoring of star-forming regions indicates that episodic
accretion may occur even earlier, at the Class 0 stage (Safron et al., 2015;
Johnstone et al., 2018).

The physical mechanisms responsible for the outburst remain a matter of
debate. A funneling mechanism in which matter from an envelope is accumulated
in the accretion disk and released episodically could explain the outbursts
(Zhu et al., 2009). Thermal instabilities can also be responsible for the
piling up and sudden release of inner disk material (Bell & Lin, 1994).
Additionally, the formation and fragmentation of spiral waves, driven by
gravitational instabilities, have been suggested to trigger these accretion
outbursts (see Kratter & Lodato, 2016, for a review). Observations of spiral
structures due to gravitational instabilities, such as those seen in the
massive disk of Elias 2-27 (Pérez et al., 2016) and the recently observed
fragmenting spirals around the V960 Mon system after its FUor-like outburst in
2014 (Weber et al., 2023), support this theory.

Despite their significance in our understanding of low-mass star formation,
the exact causes of these outbursts remain elusive. The dense star-forming
environments of these young circumstellar disks offer another possible
explanation. It is proposed that the sudden outbursts could be linked to
interactions within binary systems or encounters with passing stars, which are
quite common in these crowded regions (Bonnell & Bastien, 1992; Reipurth &
Aspin, 2004). Additionally, the multiplicity of stars and the substantial
cross-sectional area of these disks can foster interactions with passing stars
(star-disk flybys; Pfalzner, 2008; Cuello et al., 2019) as well as disk-disk
interactions which can excite the disks turning them unstable (Muñoz et al.,
2015).

Various FUor-like protostars have been identified as binary companions,
including notable cases such as L1551 IRS5, RNO 1B/C, AR 6A/B, HBC 494 (Pueyo
et al., 2012; Nogueira et al., 2023, and references therein). The archetypical
FU Ori system, FU Orionis itself, is a known binary composed of two stars
separated by 0.5″ where the northern component (FU Ori N hereafter) is the
optically visible star driving the outburst. Wang et al. (2004) discovered the
southern binary component, which according to a spectroscopical analysis by
Beck & Aspin (2012), is the more massive of the two (hence FU Ori S is the
primary member). Early ALMA Observations of FU Ori at 0.5″ resolution resolved
the binary system in continuum and revealed complex kinematical features
(Hales et al., 2015). High-resolution ALMA observations of FU Ori at
0.04″resolution by Pérez et al. (2020) resolved each individual disk,
reporting very compact sizes possibly truncated by binary interactions, and
kinematical signatures of Keplerian rotation around the outbursting source (FU
Ori N). These discoveries contribute to the ongoing debate on whether the
enhanced accretion can be attributed to close companion interactions. Another
perspective posits perturbation by sub-stellar companions (Clarke et al.,
2005).

In terms of the evolutionary stage, FUor objects seem to resemble Class I
protostars more than Class II objects. Many FUors have Class I type SEDs
(e.g., Gramajo et al., 2014) as well as dust silicate absorption features, and
therefore have been associated with Class I protostars (Quanz et al., 2006).
Most FUors are surrounded by reflection nebulae, remnants from their parent
cores, some of which are seen to brighten up during the outburst (Herbig,
1966; Goodrich, 1987; Aspin & Reipurth, 2003). FU Ori itself, located near the
Lynds Bright Nebula 878 (LBN 878), has a fan-shaped reflection nebula that
appeared next to the star after the flare, which was not present before
(Wachmann, 1954). Figure 1 a shows large-scale view of the FU Ori environment,
as seen in optical wavelengths. The optical composite image is courtesy of
astrophotographer Jim Thommes111https://jthommes.com/Astro/LBN878.htm, also
known for the discovery of the Thommes Nebula, a reflection nebula that
appeared after the outburst of FUor-type object V900 Mon (Thommes et al.,
2011).

The enormous energy released during the outburst can power winds and jets
which in turn may help disperse the primordial core, a process that will
eventually regulate the final stellar mass (Shu et al., 1987). FUors power
prominent winds measured via the presence of P Cygni profiles in some specific
lines (Connelley & Reipurth, 2018), from which the inferred mass loss rates
correspond to 10% of mass accretion rate, thus several orders of magnitude
larger than those of classical T Tauri stars (Calvet, 2004). The observed
profiles can be well reproduced by the standard magneto-centrifugally driven
wind model with the addition of an inner disk wind (Milliner et al., 2019).

Early single-dish searches for molecular outflows in FUor objects detected
12CO(3-2) in many targets, except FU Ori (Evans et al., 1994). The advent of
ALMA enabled high-resolution interferometric spectral line observations,
providing insights into the active circumstellar environments of FUors. These
environments are characterized by conspicuous, wide-angle, and slow-moving
outflows(Ruíz-Rodríguez et al., 2017a, b; Zurlo et al., 2017).

On the other hand, EXors seem to show little outflowing activity. Outflows
have been reported around the ambiguously classified FUor/EXor sources V1647
Ori and GM Cha (Cieza et al., 2018; Principe et al., 2018; Hales et al.,
2020), and only tentatively around EX Lup (Hales et al., 2018; Rigliaco et
al., 2020). Although the sample size is small, the differences in outflow
activity between FUors and EXors suggest that the two types of objects
represent different evolutionary stages, with EXors more evolved than FUors.

There is also observational evidence that most FUor sources are surrounded by
large envelopes that are still transferring material onto the disk (Kóspál et
al., 2017a; Fehér et al., 2017). It has been proposed that the imbalance
between mass transferred from the envelope to the disk and the accretion of
disk mass onto the star may trigger the disk instability responsible for the
outburst (Bell & Lin, 1994; Kóspál et al., 2017b). Studying the larger-scale
structure of eruptive sources is thus crucial for understanding the nature of
the FUor/EXor outbursts.

Asymmetric, non-keplerian gaseous structures are now routinely discovered by
ALMA. Kinematic modeling of these structures reveals that they can be
interpreted as anisotropic infall of material from the surrounding medium into
the circumstellar disk. These structures end up following elongated patterns
for which they have been named ’accretion streamers’ (see Pineda et al., 2023,
for a review). Accretion streamers have now been detected around many Class 0
and Class I sources and are relevant for improving our understanding of low-
mass star formation. For instance, their discovery directly challenges the
traditional model of axisymmetric collapse of protostellar cores (Shu, 1977).
The role accretion streamers play in episodic accretion processes still needs
to be demonstrated; to our knowledge, no accretion streamers have been
discovered around eruptive stars yet.

In this work, we present ALMA observations of the surroundings of FU Ori at
angular scales ranging from 160 to 25000 au at 1.3mm continuum and CO
isotopologes. Section 2 provides details on the ALMA observations and data
reduction, while Section 3 presents the primary observational findings and
their analysis. These results are subsequently discussed in Section 4, and
Section 5 offers a summary of the study’s findings. Throughout the paper, we
assume a distance to FU Ori of $408\pm 3\,$pc (Gaia DR3; Bailer-Jones et al.,
2021).

##  2 Observations and Data Reduction

Observations of FU Ori (project code 2017.1.00015.S, PI: J. Williams) were
carried out between November 2017 and August 2018 using the ALMA 12-m, 7-m,
and Total Power (TP) arrays in order to obtain spatial coverage from 0.4″ to
up to a map size of 1′ (160 to 25000 au). The log of the observations,
calibrators used, weather conditions, and array characteristics are shown in
Table 1. The full map required a 27-pointing mosaic for the 12-m array and a
7-pointing mosaic for the 7-m. The total antenna number for each 12-m, 7-m,
and TP arrays were 45, 11, and 3, respectively.

The correlators were configured in Frequency Division Mode (FDM) to observe
12CO(2–1), 13CO(2–1), and C18O(2–1) in a single Band 6 spectral setting,
providing spectral resolutions of 141 kHz for 12CO(2–1) and 122.070 kHz for
13CO(2–1) and C18O(2–1) corresponding to velocity resolutions of 0.184, 0.166
and 0.167 km s-1, respectively. The bandwidth coverage was 142.4, 79.7, and
80.0 km s-1 for 12CO(2–1), 13CO(2–1), and C18O(2–1). A 1.875 GHz-wide spectral
window centered at 232.470 GHz was positioned to detect the continuum dust
emission.

Table 1: Summary of ALMA Observations  Array | Date | # Pointings | UID | PWV | Flux | Phase | Baseline | Elevation | Time on | AR | MRS
---|---|---|---|---|---|---|---|---|---|---|---
|  |  |  | (mm) | Cal. | Cal. | Max-Min (m) | (deg) | Source (min) | (″) | (″)
12m | 2018-01-11 | 27 | - | 2.5 | J0510+1800 | J0532+0732 | 2516.9-15.1 | 50 | 13.6 | 0.2 | 2.7
7m | 2017-11-02 | 7 | - | 0.7 | J0522-3627 | J0532+0732 | 48.9-8.8 | 65 | 37.5 | 4.9 | 27.4
7m | 2017-11-02 | 7 | - | 0.7 | J0522-3627 | J0547+1223 | 48.9-8.8 | 55 | 37.6 | 4.9 | 27.4
TP | 2018-07-11 | - | - | 2.2 | - | - | - | 54 | 38.1 |  |
TP | 2018-07-15 | - | - | 1.5 | - | - | - | 53 | 38.1 |  |
TP | 2018-08-09 | - | - | 0.6 | - | - | - | 54 | 38.1 |  |
TP | 2018-08-09 | - | - | 0.5 | - | - | - | 54 | 38.1 |  |

All data were calibrated by the ALMA staff using the ALMA Science Pipeline
(version 40896 of Pipeline-CASA51-P2-B) in CASA 5.1.1222http://casa.nrao.edu/
(McMullin et al., 2007; CASA Team et al., 2022). The calibration process
includes offline Water Vapor Radiometer (WVR) calibration, system temperature
correction, as well as bandpass, phase, and amplitude calibrations. Online
flagging and nominal flagging such as shadowed antennas and band edges were
applied during calibration. Continuum subtraction in the visibility domain was
performed on the interferometric data prior to imaging.

Figure 1: Left: Optical RGB composite image of LBN 878 (the red and brown
nebula dominating the field) obtained by astro-photographer Jim Thommes. FU
Ori (and its reflection nebula) is the bright object located at the center of
the image. Inset shows the integrated intensity 12CO(2–1) maps as traced by
the ALMA observations (see Figure 2 for details). Redshifted and blueshifted
12CO integrated intensity maps of FU Ori are plotted over optical emission
(color scale). The blue-shifted moment 0 map (blue contours) was constructed
including emission from 8.0 to 11.5 km s-1 while the red-shifted integrated
emission (red contours) includes the emission between 12.7 and 17.5 km s-1.
The contour levels are 3, 5, and 8$\sigma$.

Imaging of the continuum and molecular emission lines was performed using the
tCLEAN task in CASA 6.1. The TP observations were converted into visibilities
using the task TP2VIS (Koda et al., 2019), and then input into tCLEAN along
with the interferometric data.

The 12-m array data was self-calibrated using the standalone version of the
automated self-calibration module (Tobin et al., in prep.). The self-
calibration process was performed separately for each observation, with one
iteration of phase-only self-calibration. The improvements in signal-to-noise
(S/N) after self-calibration ranged between 10-20%. The self-calibration
tables derived from the continuum data were applied to the FDM spectral
windows before imaging the CO lines. The continuum and spectral line data were
imaged using the tCLEAN task interactively in CASA (using a Hogbom
deconvolver, creating manually-defined CLEAN masks, and using natural
weighting to maximize sensitivity). Continuum subtraction was performed before
imaging each molecular line using the task UVCONTSUB. During cleaning of the
spectral lines, the spectral axis was binned  on-the-fly to 0.25 km s-1 to
increase the S/N. The resulting spatial resolution for the combined 12m+7m+TP
dataset corresponds to a synthesized beam size of 0.4$\arcsec$ $\times$
0.3$\arcsec$ (PA = 41.6∘).

Integrated intensity (moment 0) and intensity-weighted mean velocity (moment
1) maps for the spectral line data were produced with CASA task IMMOMENTS. The
moment 0 images were produced by integrating the signal in channels with
emission above 3$\sigma$. There is a strong cloud contamination close to the
system’s velocity (11.75 km s-1; Hales et al., 2015), particularly strong in
the 7m+TP data; thus, for display purposes, we created separate moment 0 maps
for the red-shifted and blue-shifted portions of the observed large-scale
emission.

##  3 Results

We detect 12CO, 13CO, and C18O in a variety of angular scales. The source
large-scale structure mapped by the 7-m+TP arrays dominates the final image,
overwhelming the structures traced by the 12-m array. For this reason, we also
produced continuum and spectral line cubes using the 12-m data only. This
approach allows for a more detailed examination of the specific
characteristics and features contained in each dataset.

Figure 1 (inset) shows the integrated intensity (moment 0) map for 12CO, and
Figure 2 shows the integrated intensity maps for all three CO isotopologues at
the spatial scales traced by the 7-m+TP arrays. The angular resolution of the
combined 12+7m+TP image is 0.4″$\times$0.3″. However, the maps in Figure 2
have been smoothed to 2.5″ to enhance the sensitivity to large-scale
structures (the nominal resolution of 7m-only data is 4.9 ″ whereas the
angular resolution of the 12m-only data is 0.2 ″). Figure 3 displays the 12CO
and 13CO moment 1 maps obtained by imaging the 12m-array data only. C18O was
not detected in the 12-m array data.

###  3.1 Wide, slow-moving outflows

The ALMA observations reveal the presence of large-scale, wide-angle bipolar
outflows for the first time around the class prototype FU Ori. The blue-
shifted portion of the 12CO emission traces the northeastern part of the well-
known reflection nebula. The 13CO emission partially coincides with the 12CO
emission (Figure 2), whereas the C18O is more compact. The 12CO and 13CO
emission have an extension of approximately 20000 au. The line-of-sight
velocity of the outflows is low, typically around $\sim$1-2 km s-1. The wide-
angle, slow-moving bipolar outflow traced by 12CO emission is reminiscent of
the emission detected towards other FUor objects: V883 Ori (Ruíz-Rodríguez et
al., 2017a), HBC 494 (Ruíz-Rodríguez et al., 2017a), V2775 Ori (Zurlo et al.,
2017), V1647 Ori (Principe et al., 2018), V900 Mon (Takami et al., 2019), and
GM Cha (Hales et al., 2020) as well as in low-mass Class I young stellar
objects (e.g. Arce & Sargent, 2006; González-Ruilova, C. et al.  in prep., ).
This slow outflow is also comparable to those seen towards First Hydrostatic
Core candidates (e.g. Pineda et al., 2011; Maureira et al., 2020).

Figure 2: Integrated intensity in FU Ori maps for 12CO, 13CO and C18O. The
blue-shifted moment 0 maps (blue contours) were constructed including emission
from 8.0 to 11.5 km s-1 while the red-shifted images (red contours) include
the 12.7 to 17.5 km s-1. Grayscale shows the optical image from Figure 1. The
white cross at the center of the image shows the position of FU ori N. The
angular resolution of the images (2.5″$\times$2.5″) is shown as a black circle
in the lower left.

###  3.2 Streamer in FU Ori

Figure 3 shows the zoomed-in view of the 12CO(2-1) and 13CO(2-1) integrated
velocity maps (moment 1) of FU Ori, as revealed by the 12-m array data only.
The intermediate-scale emission traced by the 12-m Array shows complex
kinematics. The 12CO(2-1) map is similar to the Cycle 0 12CO(3-2) map
presented in Hales et al. (2015), yet significantly more extended. The
elongated feature extending south of the central objects at velocities 10-11
km s-1, also observed in the Cycle 0 12CO(3-2) channel maps, is now confirmed
to be real by our deeper, higher-resolution 12CO(2-1) data. Linear structures
have been observed in scattered light around FUor type objects (Liu et al.,
2018; Takami et al., 2018), however, this is the first time such a structure
is reported in molecular line emission. Furthermore, the feature we observe in
12CO does not coincide with any of the linear features reported in the near-
IR. The elongated feature has neither connection to the larger-scale molecular
outflow nor to the inner disk rotation (Pérez et al., 2020), and is more
similar to accretion streamers recently reported around young stellar objects
(Pineda et al., 2020; Alves et al., 2019; Valdivia-Mena et al., 2022; Gupta et
al., 2023; Pineda et al., 2023).

Figure 3: Zoomed-in view of FU Ori in 12CO and 13CO emission generated imaging
the 12m-array data only. Both maps show the velocity centroids (moment 1) in
jet color scale ranging from 9.5 to 13.5 km s-1 (centered on the systemic
velocity of 11.75 km s-1). The moment zero maps are shown in contours ranging
from 3 times the zeroeth moment RMS to the peak. Contours range from 39 to 707
mJy beam-1 km s-1, and 26 to 78 mJy beam-1 km s-1, for 12CO and 13CO,
respectively. The white cross shows the position of the continuum maximum
which we take as a proxy for the position of FU Ori N. The ellipse on the
lower left represents the synthesized beam. The blue and red arrows show the
expected direction of the large-scale outflow assuming it is perpendicular to
the disk PA of 133.6∘ measured from the long baseline data in Pérez et al.
(2020).

###  3.3 Outflow properties

Swept-up cavities in the envelope of young stellar objects play a significant
role in understanding the complex processes occurring during their early
stages of formation (Arce et al., 2007, and references therein). These
cavities are created by the powerful outflows and jets emitted from
protostars, which interact with their surrounding molecular clouds. As the
outflows propagate through the dense material, they sweep away and evacuate
cavities, shaping the surrounding environment. These cavities serve as
valuable indicators of the energetic feedback mechanisms associated with star
formation. By studying the size, shape, and kinematics of these swept-out
cavities, researchers can gain insights into the dynamics of protostellar
outflows, the interaction between the outflows and their surroundings, and the
impact of these processes on the overall evolution of young stellar objects.

Assuming both the 12CO and 13CO lines trace the bipolar cavity, we use the
12CO and 13CO emissions to derive estimates of the mass of the outflow and its
kinematic properties in the traditional manner (e.g.; Cabrit & Bertout, 1990;
Dunham et al., 2014). Following the process described in Ruíz-Rodríguez et al.
(2017a), we estimate the outflow quantities from the blue- and red-shifted
emissions separately. The optically-thin tracer 13CO is used to correct for
the optical depth effects in 12CO by computing the ratio of the brightness
temperatures of each line from all the channels with detections above
5$\sigma$. To apply the correction factor to data with only 12CO detection, we
extrapolate values from a parabola fitted to the weighted mean values. Figure
4 shows the best fit parabola (solid green line). The derived outflow
properties are shown in Table 2.

Figure 4: Relationship between the brightness temperatures of 12CO (T12) and 13CO (T13) presented as a function of velocity relative to the systemic velocity. The plotted data includes representing the weighted mean values (blue solid dots), accompanied by error bars indicating the corresponding weighted standard deviations in each channel. The red solid dots depict the weighted mean values that were not employed during the fitting since the emission is optically thin in those channels and, therefore, no correction is needed. To represent the data, a green solid line is used to illustrate the best-fit second-order polynomial.  Table 2: Mass, Momentum, Luminosity, and Kinetic Energy of the Outflow Blue Lobe |  |
---|---|---
| 20 (K) | 50 (K)
Mass (M⊙) | 1.37 | 2.08
Mass loss (M⊙ yr-1) | 3.6$\times 10^{-4}$ | 5.5$\times 10^{-4}$
Momentum (M⊙ km s-1) | 1.16 | 1.76
Energy (erg) | 1.02$\times 10^{43}$ | 1.55$\times 10^{43}$
Luminosity (L⊙) | 0.02 | 0.03
Red Lobe |  |
| 20 (K) | 50 (K)
Mass (M⊙) | 0.019 | 0.029
Mass loss (M⊙ yr-1) | 5.1 $\times 10^{-6}$ | 7.7$\times 10^{-6}$
Momentum (M⊙ km s-1) | 0.014 | 0.022
Energy (erg) | 1.08$\times 10^{41}$ | 1.64$\times 10^{41}$
Luminosity (L⊙) | 2.3$\times 10^{-4}$ | 3.5 $\times 10^{-4}$

###  3.4 Streamer Model

To ascertain the infalling nature of the observed streamer, we fitted
12CO(2–1) streamer emission (see Figure 3) with analytical equations of
infalling trajectories given by Mendoza et al. (2009). For this fitting, we
use TIPSY (Trajectory of Infalling Particles in Streamers around Young stars;
Gupta et al., 2024) which simultaneously fits the morphology and velocity
profile of the streamer observations. TIPSY requires prior estimation of the
mass of the central object and systemic line-of-sight velocity of the center
of mass, which were taken to be 1.8 M⊙ and 11.75 km s-1, respectively, since
the masses of FU Ori N and FU Ori S are 0.6 and 1.2 M⊙ respectively. (Pérez et
al., 2020; Beck & Aspin, 2012). Pérez et al. (2020) inferred the disk dust
masses for FU Ori N and FU Ori S to be $22$ M⊕ and 8 M⊕, respectively.
Assuming the typical gas-to-dust ratio of 100, the total masses of the disks
are expected to be $\sim 7$ M${}_{\text{jupiter}}$ (or $\sim 6.6\times
10^{-3}$ M⊙) and $\sim 3$ M${}_{\text{jupiter}}$ (or $\sim 2.8\times 10^{-3}$
M⊙), respectively, which are negligible for the streamer model calculations.

Figure 5: The best fit infalling trajectory from TIPSY (green curved line)
overplotted on the 12CO integrated intensity map (left panel) and position-
velocity (PV) diagram along the streamer (right panel). In the PV diagram, the
outermost contours represent the noise level of the data and each subsequent
contour denote an increment by a factor of two. The red rectangles in both
panels represent the initial R.A., Decl., and VLSR limits given to TIPSY to
extract the sub-cube of the streamer emission. A clustering algorithm does the
rest of fine-tuning of the extraction (see Gupta et al., 2024, for details).

The fitting results suggest that the morphology and the velocity profile of
observed streamer emission can be well-represented as a trail of infalling
gas, as shown in Figure 5. Using the distribution of trajectories which can
fit the observed streamer, infalling timescale, of the streamer was estimated
to be $8537\pm 3317$ year. Here the infalling timescale is defined as the time
taken for infalling material to reach the closest point to the protostars in
their trajectory, starting from the farthest point in the observed streamer.
The uncertainties were computed as the standard deviation of infalling
timescales, as estimated for the trajectories that could fit the data
reasonably well (see Sec. 2.3. in Gupta et al., 2024). The uncertainty on our
trajectory solution is mainly due to lack of curvature in the projected
streamer morphology, which is useful in constraining gas velocities along the
plane of the sky. Note that the theoretical trajectories used for the fitting
assume that the only force acting on the particle is gravity from a point
source. Although FU Ori is a binary system, this assumption is still valid at
the length scale of the streamer which is roughly six times the binary
separation.

The streamer was also observed in 12CO(3–2) emission (Hales et al., 2015),
which allows us to simultaneously estimate temperature and column density of
the gas in the streamer. We estimated the brightness temperature of infalling
gas to be $11.3$ K and $7.1$ K in 12CO(2–1) and (3–2) emission, respectively.
Using the radiative transfer code, RADEX, brightness temperatures were found
to correspond to the gas temperature of $\sim 25$ K and gas column density of
$\sim 2.5\times 10^{16}$ cm-2. This is also in agreement with the non-
detection of the streamer in 13CO(2–1) emission. Assuming the typical CO to H2
ratio of $10^{-4}$, the mass of gas in the streamer is then estimated to be
$\sim 5.6\times 10^{-7}$ M⊙. Although we use the same boundaries to select
streamer emission in 2–1 and 3–2 emission maps, the resulting ratios may be
affected by the different angular scales traced by the two datasets (which
differ by a factor of 2 in terms of angular resolution and maximum recoverable
scale), thus adding extra uncertainty to the derived quantities.

Although these estimates do not include parts of streamers beyond the primary
beams of the interferometric observations, they can still be used to estimate
mass infall rates as $\dot{M}_{inf}=M_{streamer}/T_{inf}$, where
$M_{streamer}$ is observed streamer mass and $T_{inf}$ infall time for the
observed streamer. We find the streamer to be feeding the FU Ori system at a
rate of $\sim 6.6\times 10^{-11}$ M⊙ yr-1. Note that this value should be
treated as an order of magnitude estimate, because of uncertainties on the
streamer mass. A more reliable mass infall rate estimation will require
modeling the total reservoir of bound gas and ideally, in a more optically
thin line emission like 13CO(2–1), which was not detected in our streamer
observations.

##  4 Discussion

###  4.1 Molecular Outflow

Our new ALMA observations detect a wide-angle outflow emerging from the FU Ori
binary system for the first time. The northern component of the binary system,
FU Ori N, is the outbursting source. Thus it is natural to assume that FU Ori
N was the source driving the outflows. Prior searches for molecular outflows
around FUors, mainly using single-dish telescopes (Evans et al., 1994),
reported outflowing material from many FUors but failed to detect flows
emerging from the FUor class prototype. These non-detections instigated the
belief that there were no molecular outflows around the FU Ori system. Our
discovery ends the mystery by clearly demonstrating the presence of a
molecular outflow from FU Ori itself.

The main characteristics of the outflow discovered around FU Ori are its wide
angle and low velocity. The blue-shifted lobe extends approximately 10000 au
in each direction. We compute the dynamical timescale of the outflow by
dividing the extension of the blue-shifted lobe by the maximum gas velocity
(not considering correction for the system’s inclination). We obtain a
Kinematic Age of 3800 yr, which is significantly larger than the typical
duration of FUor outbursts ($<10^{2}$ yr). This implies that the current
outburst is not responsible for the observed outflows, similar to what it was
concluded for V883 Ori (Ruíz-Rodríguez et al., 2017b).

The blue-shifted lobe has a projected opening angle of $\sim$140-150∘ . Wide-
angle outflows have been reported around other FUors, such as HBC 494, V883
Ori, and GM Cha, as well as in Class I objects such as RNO 129 (e.g.; Ruíz-
Rodríguez et al., 2017a, b; Hales et al., 2020; Arce & Sargent, 2006). These
outflows are wider than those of other FUor observed with ALMA such as V346
Nor, Haro 5a IRS, V1647 Ori, V2775 Ori and V900 Mon (Kóspál et al., 2017b;
Kospal, 2018; Principe et al., 2018; Zurlo et al., 2017; Takami et al., 2019).
Arce & Sargent (2006) reported a trend in which outflow opening angles
increase with the age of the protostars. A possible explanation for the
widening of outflows with age is that in the early stages, the main observed
outflow structures are highly collimated, fast-moving jets that begin to
pierce the envelope. As most of the envelope material along the jet axis is
cleared out, the prevailing structure in the molecular outflow will be
primarily shaped by the gas carried along by the wide-angle portion of the
wind. Alternatively, the parabolic morphology of FU Ori’s wide opening
molecular outflow is consistent with the shell disk wind model (Lee et al.,
2000; Arce et al., 2007; de Valon et al., 2022).

In this context, the FU Ori system is more evolved than other Class I
protostars that are still heavily embedded in their surrounding envelope and
have narrower cavity opening angles. This interpretation is consistent with
the fact that FU Ori is optically visible because most of the surrounding
material has been accreted or swept up by mass dissipation processes (e.g.
primordial high-velocity jets, disk winds, disk+star accretion). The 12CO and
13CO outflows detected by ALMA coincide with the wide-angle conical reflection
nebula seen in optical wavelengths (Figure 1), suggesting that the dissipative
processes acting in FU Ori effectively disperse gas and dust from the
primordial envelope. Nevertheless, there is still a significant amount of
cloud/envelope material, evidenced by the substantial contamination in
systemic velocities (Figures 6-10) and the centrally peaked C18O emission
traced by the 7m+TP array observations (Figure 2), likely associated with the
remnant of a primordial envelope.

The projected velocity of the outflows is small, with a characteristic
velocity of $\sim$1-2 km s-1. Using relative velocities of 1 km s-1 and a
distance from the center of 10000 au, we find the outflow material to be
unbound. The low outflow velocity of the 12CO and 13CO outflows is much
smaller than the typical 10-40 km s-1 observed around other FUors (Ruíz-
Rodríguez et al., 2017b, and references therein) but is similar to that of
V883 Ori (0.65 km s-1 Ruíz-Rodríguez et al., 2017b). Pérez et al. (2020)
estimated an inclination 37∘ degrees for the dust disk around FU Ori N. Thus,
assuming that the outflow’s direction coincides with the disk’s angular
momentum, we discard that the low velocity observed is due to inclination
effects. Ruíz-Rodríguez et al. (2017b) proposed that in the case of V883 Ori,
the low outflow velocities can be explained if the outflow we see is very old,
a remnant of older fast-moving outflows that have now ceased or emerging from
a quiescent disk without the creation of high-velocity components, similar to
what we observe in the FU Ori system. FU Ori is known to have deep, broad
absorption P Cygni profiles that can be described by disk wind models capable
of sustaining mass-loss rates of the order of 10% the accretion rate (i.e.
mass loss due to wind of $\sim 10^{-5}$ to $10^{-6}$ M⊙ yr-1; Milliner et al.,
2019).

The properties of the outflow listed in Table 2 can be readily compared with
those of V883 Ori and HBC 494, as they were derived using the same method. The
estimated mass for the blue-shifted lobe is two orders of magnitude larger
than the red-shifted lobe. We cannot confirm whether this difference is real
or possibly caused by the absorption of cloud/envelope material in the line of
sight. Nevertheless, the mass estimation of the blue-shifted lobe is
comparable to that of V883 Ori and one order of magnitude larger than that of
HBC 494. The mass loss, momentum, energy, and luminosity are also similar to
those of V883 Ori yet still consistent with the values detected around other
young protostars (e.g. Arce & Sargent, 2006).

###  4.2 Accretion Streamer

As discussed in Section 3.4, the 12CO streamer (Figure 5) seems to be feeding
mass to the protostellar system at a rate of $\sim 6.6\times 10^{-11}$ M⊙ yr-1
(or $\sim 0.07$ M${}_{\text{jupiter}}$ Myr-1). As mentioned in Section 3.4,
the disk masses for FU Ori N and FU Ori S are $\sim 6.6\times 10^{-3}$ M⊙ and
$\sim 2.8\times 10^{-3}$ M⊙, respectively. This would suggest that the
observed streamer will require $\sim 100$ Myr to replenish disks masses, which
is at least an order of magnitude greater than the typical disk lifetimes.

The estimated mass infall rate in the streamer around FU Ori is lower than
those detected in streamers around other Class I sources ($\sim 10^{-6}$
M${}_{\odot}\,$yr-1, e.g., Valdivia-Mena et al., 2022; Pineda et al., 2023).
The streamer was not detected in C18O, consistent with it not being as massive
as the ones detected around other Class I protostars. There is also no clear
detection of streamer in 13CO emission, although there is still some hint of
stream-like emission in some 13CO channel (channels 11–11.25 km s-1, see
Figure 10), just adjacent to 12CO streamer emission (mostly in channels
10.25–10.75 km s-1, see Figure 9). Therefore, some of the 12CO streamer
emission may be hidden due to high optical depth effects. This may suggest
that we are underestimating streamer mass, although not by orders of magnitude
as we do not detect the streamer in 13CO or C18O emission.

The streamer needs to be more massive to sustain FU Ori’s outburst accretion
rates (by several orders of magnitude). The estimated streamer mass infall
rate is not even sufficiently massive to sustain quiescent stellar accretion
rates. Despite not being massive enough to sustain the outburst accretion
rate, the streamer can deliver material to the disk and trigger disk
instabilities, which can further lead to accretion. Anisotropic infall,
cloudlet capture events, inhomogeneous delivery of material, and building-up
of material around dust traps can all lead to the disk instabilities that
could trigger accretion outbursts (Vorobyov & Basu, 2010; Dullemond et al.,
2019; Kuznetsova et al., 2022; Hanawa et al., 2024). It still needs to be
determined if the low observed mass infall rate is enough to trigger disk
instabilities, for which detailed modeling would be required.

It is also possible that the streamer we observe now is not directly connected
to the current outburst but instead is a remnant of a more massive streamer
that could have played a role in feeding mass to the disk in the past, but now
has already relinquished most of its mass into the protostellar system. In
this scenario, the streamer we are observing is the smoking gun for previous,
vigorous episodes of cloud accretion into the disk that may have led the disk
to become unstable and trigger FU Ori’s outburst.

There may also be more than one streamer interacting with the binary system.
For instance, the high-resolution ALMA observations of FU Ori from Pérez et
al. (2020) detected CO emission that delineates the position of the brightest
feature in scattered light: a prominent arc to the northeast. This arc could
represent a signpost of active accretion into the system or could be light
reflected from material at the base of the outflow (Zurlo et al. in
preparation), similar to features recently reported in other outbursting
systems (Weber et al., 2023).

##  5 Conclusions

The current study presents ALMA observations of the FU Ori outbursting system
covering spatial scales from 160 to 25,000 astronomical units.

The remarkable sensitivity to a wide range of spatial scales unveil a wide,
slow-moving molecular outflow. To obtain the first determination of the
outflow properties, including mass, momentum, and kinetic energy, we have
employed 13CO(2–1) data to correct for optical depth effects.

A thorough analysis of the higher-resolution interferometric data reveals
intricate gas kinematics, likely resulting from a confluence of ongoing
physical processes, including accretion, rotation, ejection, and spreading of
material due to binary interaction

The new observations corroborate the existence of an elongated 12CO feature
that can be modeled consistently as an accretion streamer. The streamer
analysis suggests that the stream cannot sustain the enhanced accretion rate.
Nevertheless, the observed streamer may be feeding the disk, helping to
produce conditions of instability that can give rise to the outburst. Follow-
up hydro-dynamical modeling is required to ascertain the plausibility of this
scenario. Alternatively, the observed streamer might not be directly tied to
the current outburst but instead be a leftover from a prior, more massive
streamer that previously contributed to the disk mass, rendering the disk
unstable and triggering the FU Ori outburst. These results demonstrate the
value of multi-scale interferometric observations to enhance our understanding
of the FU Ori outbursting system and provide new insights into the complex
interplay of physical mechanisms governing the behavior of FUor-type and the
many kinds of outbursting stars.

## Acknowledgments

We express our gratitude to Jim Thommes for generously providing the optical
image of FU Orionis. This paper makes use of the following ALMA data:
ADS/JAO.ALMA#2017.1.00015.S. ALMA is a partnership of ESO (representing its
member states), NSF (USA) and NINS (Japan), together with NRC (Canada) and NSC
and ASIAA (Taiwan), in cooperation with the Republic of Chile. The Joint ALMA
Observatory is operated by ESO, AUI/NRAO and NAOJ. The National Radio
Astronomy Observatory is a facility of the National Science Foundation
operated under a cooperative agreement by Associated Universities, Inc. This
work acknowledges support from ANID – Millennium Science Initiative Program –
Center Code NCN2021_080. S.P. acknowledges support from FONDECYT grant
1231663. A.S.M. acknowledges support from ANID / Fondo 2022 ALMA / 31220025.
P.W. acknowledges support from FONDECYT grant 3220399. J. P. W. acknowledges
support from NSF grant AST-2107841.

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##  Appendix A Channel maps

Figures 6, 7, and 8 show the 12m+7m+TP array channel maps for 12CO(2–1),
13CO(2–1), and C18O(2–1), respectively.

Figure 6: 12CO channel maps towards FU Ori (7m+TP, smoothed to 2.5”
resolution). The velocity of the channels is shown in the Local Standard of
Rest (LSR) frame, centered at the rest frequency of 12CO(2–1). The data has
been binned to a velocity resolution of 0.25 km s-1. Contours of the 1.3mm
continuum emission at 0.3” resolution are overlaid (contour levels are 0.5, 1,
and 1.8 mJy beam-1). Figure 7: Same as Figure 6 but for 13CO. Figure 8: Same
as Figure 6 but for C18O.  Figure 9: 12CO FU Ori maps for array C43-5
configuration. Contours of the 1.3mm continuum emission at 0.3” resolution are
overlaid in green (contour levels are 2 and 6 mJy beam-1). Figure 10: 13CO FU
Ori maps for array C43-5 configuration. Contours of the 1.3mm continuum
emission at 0.3” resolution are overlaid (contour levels are 2 and 6 mJy
beam-1).