# Crossing lemma for the odd-crossing number

János Karl
Budapest University of Technology and Economics
<EMAIL_ADDRESS>
and
Géza Tóth
Alfréd Rényi Institute of Mathematics, Budapest and Budapest University of
Technology and Economics
<EMAIL_ADDRESS>Supported by National Research, Development and Innovation
Office, NKFIH, K-131529 and ERC Advanced Grant ”GeoScape” 882971.

###### Abstract

A graph is $1$-planar, if it can be drawn in the plane such that there is at
most one crossing on every edge. It is known, that $1$-planar graphs have at
most $4n-8$ edges.

We prove the following odd-even generalization. If a graph can be drawn in the
plane such that every edge is crossed by at most one other edge an odd number
of times, then it is called 1-odd-planar and it has at most $5n-9$ edges. As a
consequence, we improve the constant in the Crossing Lemma for the odd-
crossing number, if adjacent edges cross an even number of times. We also give
upper bound for the number of edges of $k$-odd-planar graphs.

##  1 Introduction

By a graph we always mean a simple graph, that is, a graph with no loops and
no parallel edges. We use the term multigraph if loops and parallel edges are
allowed. A drawing of a (multi)graph in the plane is a representation such
that vertices are represented by distinct points and its edges by curves
connecting the corresponding points. We assume that no edge passes through any
vertex other than its endpoints, no two edges touch each other (i.e., if two
edges have a common interior point, then at this point they properly cross
each other), no three edges cross at the same point, and two edges cross only
finitely many times.

The crossing number of a graph $G$, ${\mbox{\sc cr}}(G)$, is the minimum
number of crossings (crossing points) over all drawings of $G$. The pair-
crossing number, ${\mbox{\sc pcr}}(G)$, is the minimum number of pairs of
crossing edges over all drawings of $G$. In an optimal drawing for ${\mbox{\sc
cr}}(G)$, any two edges cross at most once [S04]. Therefore, it is not easy to
see the difference between these two definitions. Indeed, there was some
confusion in the literature between these two notions, until the systematic
study of their relationship [PT00a]. Clearly, ${\mbox{\sc
pcr}}(G)\leq{\mbox{\sc cr}}(G)$, and in fact, we cannot rule out the
possibility, that ${\mbox{\sc cr}}(G)={\mbox{\sc pcr}}(G)$ for every graph
$G$. Probably it is the most interesting open problem in this area. From the
other direction, the best known bound is ${\mbox{\sc cr}}(G)=O({\mbox{\sc
pcr}}(G)^{3/2}\log{\mbox{\sc pcr}}(G))$ [S17], [KT21].

The odd-crossing number, ${\mbox{\sc ocr}}(G)$, is the minimum number of pairs
of edges that cross an odd number of times, over all drawings of $G$. Clearly,
(as non-crossing edges cross an even number of times) for every graph $G$,
${\mbox{\sc ocr}}(G)\leq{\mbox{\sc pcr}}(G)\leq{\mbox{\sc cr}}(G)$. According
to the (weak) Hanani-Tutte theorem [C34], [PSS07], if ${\mbox{\sc ocr}}(G)=0$,
then $G$ is planar, that is, ${\mbox{\sc ocr}}(G)={\mbox{\sc
pcr}}(G)={\mbox{\sc cr}}(G)=0$. It was shown in [PSS07] that for $k=1,2,3$, if
${\mbox{\sc ocr}}(G)=k$, then ${\mbox{\sc ocr}}(G)={\mbox{\sc
pcr}}(G)={\mbox{\sc cr}}(G)=k$. There are examples where ocr is different from
pcr and cr, namely, there is an infinite family of graphs with ${\mbox{\sc
ocr}}(G)<0.855\cdot{\mbox{\sc pcr}}(G)$ [T08], [PSS08]. From the other
direction we only have ${\mbox{\sc pcr}}(G)<2{\mbox{\sc ocr}}(G)^{2}$ [PT00a].

In [PT00b] some further variants were introduced, in order to study the role
of crossings between adjacent edges. For each of cr, pcr, and ocr, they
introduced three counting rules.

Rule $+$: Only those drawings are considered, where adjacent edges cannot
cross.

Rule $0$: Adjacent edges can cross and their crossings are counted as well.

Rule $-$: Adjacent edges can cross and their crossings are not counted.

Combining these rules with the three crossing numbers, we get nine
possibilities. But it is easy to see that ${\mbox{\sc cr}}_{+}={\mbox{\sc
cr}}$ [PT00b]. On the other hand, regarding Rule $+$ for the odd-crossing
number, it seems more natural to assume that adjacent edges cross an even
number of times than to assume that they do not cross at all. So, let
${\mbox{\sc ocr}}_{*}(G)$ be the minimum number of odd-crossing pairs of edges
over all drawings of $G$ where adjacent edges cross an even number of times
(these drawings are called weakly semisimple in [BFK15]). Therefore, we have
nine versions, see Table 1. In this table, values do not decrease if we move
to the right or up, and it was shown in [PSS08] that ${\mbox{\sc
cr}}(G)<2{\mbox{\sc ocr}}_{-}(G)^{2}$. On the other hand, there are graphs $G$
such that ${\mbox{\sc ocr}}_{-}(G)<{\mbox{\sc ocr}}(G)$ [FPSS11].

Rule $+$ | ${\mbox{\sc ocr}}_{*}(G)\leq{\mbox{\sc ocr}}_{+}(G)$ | ${\mbox{\sc pcr}}_{+}(G)$ | ${\mbox{\sc cr}}(G)$
---|---|---|---
Rule $0$ | ${\mbox{\sc ocr}}(G)$ | ${\mbox{\sc pcr}}(G)$
Rule $-$ | ${\mbox{\sc ocr}}_{-}(G)$ | ${\mbox{\sc pcr}}_{-}(G)$ | ${\mbox{\sc cr}}_{-}(G)$

Table 1. Nine versions of the crossing number.

The Crossing Lemma, discovered by Ajtai, Chvátal, Newborn, Szemerédi [ACNS82]
and independently by Leighton [L84] is definitely the most important
inequality for crossing numbers.

Crossing Lemma. If a simple graph $G$ of $n$ vertices has $m\geq 4.5n$ edges,
then ${\mbox{\sc cr}}(G)\geq\frac{1}{60.75}\frac{m^{3}}{n^{2}}$ edges.

The bound is tight, apart from the value of the constant [PT97]. The constant
above follows from the beautiful probabilistic argument of Chazelle, Sharir
and Welzl [AZ04]. This argument works for all nine versions of the crossing
number [PT00b]. For the classical crossing number, ${\mbox{\sc cr}}(G)$, the
constant was improved in three steps [PT97], [PRTT06], the best bound is due
to Ackerman [A19], ${\mbox{\sc cr}}(G)\geq\frac{1}{29}\frac{m^{3}}{n^{2}}$,
when $m\geq 7n$.

The only improvement for any other version is a result of Ackerman and
Schaefer [AS14], ${\mbox{\sc
pcr}}_{+}(G)\geq\frac{1}{34.2}\frac{m^{3}}{n^{2}}$, when $m\geq 6.75n$. For
all other versions of the crossing number, the constant $60.75$ is the best we
have.

In this note we get an improvement for two other versions, ${\mbox{\sc
ocr}}_{+}$ and ${\mbox{\sc ocr}}_{*}$.

######  Theorem 1.

Suppose that $G$ has $n$ vertices and $m\geq 6n$ edges. Then ${\mbox{\sc
ocr}}_{+}(G)\geq{\mbox{\sc ocr}}_{*}(G)\geq\frac{1}{54}\frac{m^{3}}{n^{2}}$.

Our approach is very similar to all previous improvements mentioned above. The
first step is to find many odd-crossing pairs in sparse graphs. Then this
bound is applied for a random subgraph of $G$ to get the general bound.

A graph $G$ is called $k$-planar if it can be drawn in the plane such that
there are at most $k$ crossings on each edge. Such a drawing is called a
$k$-plane drawing. Let $m_{k}(n)$ denote the maximum number of edges of a
$k$-planar graph of $n$ vertices.

Clearly, $m_{0}(n)=3n-6$. It is known that $m_{1}(n)=4n-8$ for $n\geq 12$,
$m_{2}(n)\leq 5n-10$ and it is tight for infinitely many values of $n$,
[PT97], $m_{3}(n)\leq 5.5n-11$, $m_{4}(n)\leq 6n-12$, which are tight up to an
additive constant [PRTT06], [A19].

We prove an odd-even version of these results. A graph $G$ is called $k$-odd-
planar if it can be drawn in the plane such that any edge is crossed an odd
number of times by at most $k$ other edges (edges crossing an even number of
times do not count). Such a drawing is called a $k$-odd-plane drawing.

Let $m^{\mathrm{odd}}_{k}(n)$ denote the maximum number of edges of a $k$-odd-
planar graph with $n$ vertices. Clearly, we have $m^{\mathrm{odd}}_{k}(n)\geq
m_{k}(n)$ and by the weak Hanani-Tutte theorem [C34], [PSS07], we have
$m^{\mathrm{odd}}_{0}(n)=3n-6$.

######  Theorem 2.

For any $n,k\geq 1$ we have

$m^{\mathrm{odd}}_{k}(n)\leq m_{k}(n)+k(n-1).$

This result is interesing only for small $k$. For $k$ (and $n$) large enough,
we have an easy better bound.

######  Theorem 3.

For any $n,k\geq 1$ we have

$m^{\mathrm{odd}}_{k}(n)\leq\sqrt{32}\sqrt{k}n.$

We do not think that our bounds are tight. We cannot even rule out the
possibility that $m^{\mathrm{odd}}_{k}(n)=m_{k}(n)$ for every $n,k$.

##  2 Proofs

A (multi)graph $G$, together with its drawing $D$ in the plane, is called
topological (multi)graph. The points (resp. curves), representing the vertices
(resp. edges) of $G$ are called vertices (resp. edges) of the topological
(multi)graph. If the drawing is obvious from the context, we do not make any
notational distinction between the topological (multi)graph and the underlying
abstract (multi)graph. Let $G$ be a topological multigraph and $e$ an edge.
The pieces of $e$ in small neighborhoods of its endpoints are called endings
of $e$ and denoted by $e^{+}$ and $e^{-}$. As $e$ is an undirected edge, the
$+$ and $-$ signs have no special meaning, either ending can be $e^{+}$ and
the other one is $e^{-}$. The rotation system is the cyclic order of adjacent
edges, or endings, at each vertex. A cyclic order is always clockwise. Two
edges form an odd pair (resp. even pair) if they cross an odd (resp. even)
number of times. An edge is called even if it is crossed an even number of
times by every other edge and it is odd otherwise.

According to the weak Hanani-Tutte theorem, if a graph can be drawn so that
any two edges cross an even number of times, then it is planar. This result
has many proofs, one of the nicest and simplest is due to Pelsmajer, Schaefer
and Štefankovič [PSS07]. The proof is based on the following lemma.

Lemma 0. [PSS07] Let $G$ be a topological multigraph that has one vertex and
$n$ edges (loops). Suppose that every edge is even. Then, $G$ can be redrawn
such that the rotation system is the same and there is no edge crossing.

First we prove the following generalization.

######  Lemma 1.

Let $G$ be a topological multigraph that has one vertex and $m$ edges (loops).
Then, $G$ can be redrawn such that (i) the rotation system is the same (ii)
even pairs do not cross, (iii) odd pairs cross once, and (iv) there are no
self-crossings.

Proof. The proof is by induction on the number of edges. If there is only one
loop, the statement is trivial.

Suppose that $G$ has one vertex $v$ and $m>1$ loops, and the statement has
been proved for a smaller number of loops. Let $D$ denote the present drawing
of $G$. We can assume immediately that there are no self-crossings, since we
can eliminate self-crossings of each loop without changing the number of
crossings between different loops.

Let $s$ be a very short segment ending in $v$ and not crossing any of the
loops.

Assume without loss of generality that for any loop $e$, $s$, $e^{-}$ and
$e^{+}$ are in this clockwise order at $v$ (otherwise we can switch the
notation for $e^{-}$ and $e^{+}$).

For two loops $e$ and $f$, we say that $f\prec e$ if the clockwise order of
endings at $v$ is $s$, $e^{-}$, $f^{-}$, $f^{+}$, $e^{+}$. The relation
$\prec$ defines a partial order on the loops. Let $e$ be a minimal loop with
respect to $\prec$. That is, any other loop $f$ has at most one ending between
$e^{-}$ and $e^{+}$.

Delete $e$ and apply the induction hypothesis. We get a drawing $D^{\prime}$
satisfying the conditions. In particular, the cyclic order of endings is the
same as in $D$. Insert $e^{-}$ and $e^{+}$ back to their original places and
connect them by an arc in a small neighborhood of $v$, going clockwise from
$e^{-}$ to $e^{+}$. We can do it such that it crosses only those endings, each
exactly once, which are between $e^{-}$ and $e^{+}$. In the obtained drawing
$D_{1}$, the rotation system is the same as in $D$ and there are no self-
crossings. For any two loops $f,g\neq e$, (ii) and (iii) are satisfied by the
induction hypothesis. If $e,f$ is an odd pair in $D$, then it has exactly one
ending between $e^{-}$ and $e^{+}$ so in $D_{1}$ they cross once. If it is an
even pair in $D$, then $f$ has no ending between $e^{-}$ and $e^{+}$ so in
$D_{1}$ they do not cross. This concludes the proof of Lemma 1. See Figure 1.
$\Box$

Figure 1: Steps in the proof of Lemma 1.

Remarks. 1\. The statement of Lemma 1 can be found implicitly in [PSS07], p.
492, as a remark.

2\. Another possible proof is the following. Take a drawing of $G$ which has
the same rotation system and under this condition the minimum number of
crossings. It can be shown that this drawing satisfies the conditions,
otherwise we could have a better drawing of $G$. But we did not find this
method easier.

######  Lemma 2.

Let $k\geq 0$, $l\geq 1$. Suppose that $n_{1},\ldots,n_{l}>0$ and
$n_{1}+\cdots+n_{l}=n$. Then

$m_{k}(n)\geq m_{k}(n_{1})+\cdots+m_{k}(n_{l}).$

Proof. For every $i$ take a $k$-plane graph of $n_{i}$ vertices and
$m_{k}(n_{i})$ edges. Their disjoint union is a $k$-plane graph with $n$
vertices and $m_{k}(n_{1})+\cdots+m_{k}(n_{l})$ edges. $\Box$

Definition. Suppose that $G$ is a topological multigraph. Let $e=uv$ be an
edge. The contraction of $e$ to $u$ is the following procedure. Move $v$ to
$u$, along the edge $e$. At the same time, extend all other edges incident to
$v$ by curves along $e$, from a neighborhood of $v$ to $u$, without creating
any crossing among them. See Figure 2.

Figure 2: Contracting edge $e=uv$ to $u$. The resulting vertex is called $w$.

Suppose that $e=uv$ is an even edge of $G$. Contract $e$ to $u$ and let $w$ be
the new vertex and $G^{\prime}$ the resulting topological graph. There is a
natural bijection between the edges of $G\setminus e$ and $G^{\prime}$, and
the odd pairs are exactly the same. Any vertex $x\neq u,v$ of $G$ remains a
vertex of $G^{\prime}$ with the same cyclic order of incident edges. Suppose
that in $G$, $e,e_{1},\ldots,e_{a}$ and $e,f_{1},\ldots,f_{b}$ are the cyclic
orders of incident edges at $u$ and $v$, respectively. Then in $G^{\prime}$,
the cyclic order of edges at $w$ is $e_{1},\ldots,e_{a},f_{1},\ldots,f_{b}$.
We can “reverse” the contraction as follows. If we replace $w$ in $G^{\prime}$
by two very close vertices $u^{\prime}$ and $v^{\prime}$, connected by an
edge, $u^{\prime}$ incident to $e_{1},\ldots,e_{a}$ and $v^{\prime}$ incident
to $f_{1},\ldots,f_{b}$, we get a topological graph with exactly the same
rotation system as $G$.

Proof of Theorem 2. Suppose that $G$ has $n$ vertices and $m$ edges, and it is
drawn in the plane such that any edge is crossed by at most $k$ other edges an
odd number of times. We can assume that $G$ is connected, as an abstract
graph, otherwise, by Lemma 2, we can argue separately for each component.
Suppose first that $G$ contains a spanning tree $F$ whose edges cross each
other an even number of times. Remove all edges of $G$ that cross an edge of
$F$ an odd number of times. We get the topological graph $G_{1}$ in the
inherited drawing. Since $F$ has $n-1$ edges and each of them is crossed by at
most $k$ other edges an odd number of times, for the number of $m_{1}$ edges
of $G_{1}$, $m_{1}\geq m-k(n-1)$. In $G_{1}$, all edges of $F$ are even.
Contract all edges of $F$. We get a topological multigraph $G_{2}$, which has
only one vertex $v$. The odd pairs are exactly the same as in $G_{1}$. Apply
Lemma 1 for $G_{2}$ and get the topological multigraph $G_{3}$. Since the
rotation system is the same in $G_{3}$ as in $G_{2}$, we can reverse the
contractions without creating any additional crossing. This way we get
$G_{4}$. Observe that $G_{4}$ is a redrawing of $G_{1}$, with the property
that if two edges form an even (odd) pair in $G$, then they do not cross
(cross once) in $G_{4}$. By the assumption, every edge in $G_{1}$ is part of
at most $k$ odd pairs. Therefore, $G_{4}$ is a $k$-plane drawing of $G_{1}$ so
it has at most $m_{k}(n)$ edges. Consequently, for the number of edges of $G$
we have $m\leq m_{k}(n)+k(n-1)$.

Suppose now that $G$ does not contain a spanning tree whose edges cross each
other an even number of times. Let $F$ be a maximal forest whose edges cross
each other an even number of times. Let $V_{1},\ldots,V_{l}$ be the vertices
of the connected components of $F$, $|V_{i}|=n_{i}$. Since $F$ is not a tree,
$l\geq 2$ and $|E(F)|=n-l$. By the maximality of $F$, those edges of $G$ that
connect two components cross some edge of $F$ an odd number of times. Remove
all edges of $G$ that cross some edge of $F$ an odd number of times. We
removed at most $k(n-l)$ edges. The resulting graph has $l$ components
$G_{1},G_{2},\ldots,G_{l}$, on vertices $V_{1},\ldots,V_{l}$. Let
$|E(G_{i})|=m_{i}$. By the construction, each $G_{i}$ contains a spanning tree
$F_{i}$ whose edges are even. Contract the edges of $F_{i}$ and argue as
above. We get that

$m\leq\sum_{i=1}^{l}m_{i}+k(n-l)\leq\sum_{i=1}^{l}m_{k}(n_{i})+k(n-l)\leq
m_{k}(n)+k(n-1).$

$\Box$

Proof of Theorem 1. We have $m^{\mathrm{odd}}_{0}(n)=3n-6$, and by Theorem 2,
$m^{\mathrm{odd}}_{1}(n)\leq m_{1}(n)+n-1=5n-9$.

First we show, by induction on the number of edges that for any graph with $n$
vertices and $m$ edges, ${\mbox{\sc ocr}}(G)\geq m-3n$. If $m\leq 3n$, the
statement is trivial. Suppose that $m>3n$ and we have proved the statement for
$m-1$. Take any drawing $D$ of $G$. Since $m^{\mathrm{odd}}_{0}(n)=3n-6$,
there is an odd pair $e,f$. Remove $e$ from $G$, and then the obtained drawing
has one less edges and at least one less odd pairs. Therefore, the number of
odd pairs in $D$ is at least $1+(m-1)-3n=m-3n$.

Now we show, again by induction on the number of edges that for any graph with
$n$ vertices and $m$ edges, ${\mbox{\sc ocr}}(G)\geq 2m-8n$. If $m\leq 5n$,
then ${\mbox{\sc ocr}}(G)\geq m-3n\geq 2m-8n$. Suppose that $m>5n$ and we have
proved the statement for $m-1$. Take any drawing $D$ of $G$, since
$m^{\mathrm{odd}}_{1}(n)\leq 5n-9$, there is an edge $e$ in two odd pairs,
$e,f$ and $e,g$. Remove $e$ from $G$, and then the obtained drawing has one
less edges and at least two less odd pairs. Therefore, the number of odd pairs
in $D$ is at least $2+2(m-1)-8n=2m-8n$.

Let $G$ be a graph of $n$ vertices and $m\geq 6n$ edges, drawn in the plane
realizing ${\mbox{\sc ocr}}_{*}(G)$, that is, any two adjacent edges cross an
even number of times and there are ${\mbox{\sc ocr}}_{*}(G)$ pairs of edges
that cross an odd number of times. Take a random subgraph $G^{\prime}$ such
that we take each vertex independently with probability $p=6n/m$. Let
$n^{\prime}$, $m^{\prime}$, and $x(G^{\prime})$ denote the number of vertices
(resp. edges) of $G^{\prime}$, and the number of odd-crossing pairs of edges
in $G^{\prime}$, in the inherited drawing. We have

$E(n^{\prime})=pn,\ \ E(m^{\prime})=p^{2}m,\ \ E({\mbox{\sc
ocr}}_{*}(G^{\prime}))\leq E(x(G^{\prime}))=p^{4}{\mbox{\sc ocr}}_{*}(G).$

For $G^{\prime}$ we have ${\mbox{\sc ocr}}_{*}(G^{\prime})\geq{\mbox{\sc
ocr}}(G^{\prime})\geq 2m^{\prime}-8n^{\prime}$, taking expected values,

$p^{4}{\mbox{\sc ocr}}_{*}(G)\geq 2p^{2}m-8pn.$

For $p=6n/m$, we have

${\mbox{\sc ocr}}_{*}(G)\geq\frac{1}{54}\frac{m^{3}}{n^{2}}.$

$\Box$

Remark. Combining Theorem 2 and the bounds for $m_{k}(n)$ we obtain that
$m^{\mathrm{odd}}_{1}(n)\leq 5n-9$ and $m^{\mathrm{odd}}_{2}(n)\leq 7n-12$. In
the proof of Theorem 1 we used only the first inequality, the second would not
help. However, if we could prove that $m^{\mathrm{odd}}_{2}(n)\leq 6.8n+c$ for
some constant $c$, then we would get an improvement in Theorem 1 as well.

Proof of Theorem 3. We apply a version of the Crossing Lemma for the odd-
crossing number, from [PT00a]: If $G$ has $n$ vertices and $m$ edges, and
$m\geq 4n$, then ${\mbox{\sc ocr}}(G)\geq\frac{1}{64}\frac{m^{3}}{n^{2}}$.

Let $k\geq 1$. Suppose that $G$ is $k$-odd-planar with $n$ vertices and
$m=m^{\mathrm{odd}}_{k}(n)$ edges. We can assume that $m\geq 4n$, otherwise we
are done.

Take a $k$-odd-plane drawing of $G$. Every edge participates in at most $k$
odd pairs. Therefore, there are at most $km/2$ odd pairs, so ${\mbox{\sc
ocr}}(G)\leq km/2$. On the other hand, we can apply the Crossing Lemma for the
odd-crossing number, so

$km/2\geq{\mbox{\sc ocr}}(G)\geq\frac{1}{64}\frac{m^{3}}{n^{2}}.$

It follows that $m\leq\sqrt{32}\sqrt{k}n$. $\Box$

Remarks. 1\. It is easy to see that Theorem 3 is better than Theorem 2 if
$n,k\geq 40$. This threshold $40$ can be improved, but it is not very
interesting.

2\. The analogue of Theorem 3 for $m_{k}(n)$ instead of
$m^{\mathrm{odd}}_{k}(n)$ was first proved in [PT97]. Using the best known
constant for the Crossing Lemma from [A19] we can get that for any $n,k\geq
2$, $m_{k}(n)\leq 3.81\sqrt{k}n$.

3\. In the proof of Theorem 1 we had to assume that adjacent edges cross an
even number of times. Therefore, it holds only for ${\mbox{\sc ocr}}_{*}(G)$
and ${\mbox{\sc ocr}}_{+}(G)$. We believe that it is not necessary and Theorem
1 can be extended to ${\mbox{\sc ocr}}(G)$. Similarly, we believe that the
bound of Ackerman and Schaefer [AS14], ${\mbox{\sc
pcr}}_{+}(G)\geq\frac{1}{34.2}\frac{m^{3}}{n^{2}}$ can be extended to
${\mbox{\sc pcr}}(G)$. We also believe that both of them are very interesing
problems.

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11institutetext: Dominion Radio Astrophysical Observatory, Herzberg Astronomy
and Astrophysics Research Centre, National Research Council Canada, P. O. Box
248, Penticton, BC V2A 6J9 Canada
11email<EMAIL_ADDRESS>22institutetext: Physics & Astronomy,
University of Victoria, Victoria, British Columbia, Canada 33institutetext:
Physics & Astronomy, University of Calgary, Calgary, Alberta, Canada
44institutetext: Dunlap Institute for Astronomy and Astrophysics University of
Toronto, Toronto, ON M5S 3H4, Canada 55institutetext: Department of Physics,
Graduate School of Science, Nagoya University, Furo-cho, Chikusa-ku, Nagoya
464-8602, Japan 66institutetext: Korea Astronomy and Space Science Institute,
776 Daedeok-daero, 34055 Daejeon, Republic of Korea 77institutetext:
Université de Paris and Université Paris Saclay, CEA, CNRS, AIM, F-91190 Gif-
sur-Yvette, France 88institutetext: Herzberg Astronomy and Astrophysics
Research Centre, National Research Council Canada, 5071 West Saanich Road,
Victoria BC V9E 2E7, Canada 99institutetext: Department of Earth Science and
Astronomy, Graduate School of Arts and Sciences, The University of Tokyo,
3-8-1 Komaba, Meguro, Tokyo 153-8902, Japan 1010institutetext: Department of
Astrophysics/IMAPP, Radboud University, P.O. Box 9010, 6500 GL Nijmegen, The
Netherlands

# Orion A’s complete 3D magnetic field morphology

M. Tahani 11 J. Glover 11 2 2 W. Lupypciw 1133 J.L. West 44 R. Kothes 11 R.
Plume 33 S. Inutsuka 55 M-Y. Lee 66 I.A. Grenier 77 L.B.G. Knee 88 J.C. Brown
33 Y. Doi 99 T. Robishaw 11 M. Haverkorn 1010

(Received Date; accepted Date)

Magnetic fields permeate the interstellar medium and are important in the star
formation process. Determining the 3D magnetic fields of molecular clouds will
allow us to better understand their role in the evolution of these clouds and
formation of stars. We fully reconstruct the approximate three-dimensional
(3D) magnetic field morphology of the Orion A molecular cloud (on scales of a
few to $\sim 100$ pc) using Galactic magnetic field models, as well as present
line-of-sight and plane-of-sky magnetic field observations. While previous
studies identified the 3D magnetic field morphology of the Orion A cloud as an
arc shape, in this study we provide the orientation of this arc-shaped field
and its plane-of-sky direction, for the first time. We find that this 3D field
is a tilted, semi-convex (from our point of view) structure and mostly points
in the direction of decreasing latitude and longitude on the plane of the sky,
from our vantage point. The previously identified bubbles and events in this
region were key in shaping this arc-shaped magnetic field morphology.

###### Key Words.:

magnetic fields, ISM: clouds, ISM: magnetic fields, stars: formation

##  1 Introduction

The role of magnetic fields on different scales of star formation (from the
formation of clouds to the formation of stars) is poorly understood. A major
limitation in this understanding is the lack of knowledge of the three-
dimensional (3D) magnetic fields in the interstellar medium (ISM). Multi-
wavelength magnetic field observations are necessary to determine the 3D
magnetic fields associated with star-forming regions.

Far-infrared dust polarized emission (e.g., Houde et al. 2004; Poidevin et al.
2011; Fissel et al. 2016; Pattle & Fissel 2019; Doi et al. 2020) and near-
infrared/optical (e.g., Pereyra & Magalhães 2004; Clemens et al. 2020)
starlight polarization observations have enabled us to probe the plane-of-sky
orientation of magnetic fields ($B_{\perp}$) in a number of molecular clouds.
Dust polarization observations revealed that magnetic field lines tend to be
perpendicular to high column density ($>10^{21.7}$ cm-2) filamentary
structures (Planck Collaboration et al. 2016a, c), allowing for greater mass
accumulations and denser filaments (e.g., Inoue et al. 2018; Hennebelle &
Inutsuka 2019).

Radio observations of Faraday rotation (Tahani et al. 2018) and Zeeman
measurements (e.g., Goodman et al. 1989; Heiles 1997; Troland & Crutcher 2008)
have been used to probe the line-of-sight magnetic fields ($B_{\parallel}$) of
molecular clouds. Tahani et al. (2018) developed a new technique based on
Faraday rotation measurements for determining the strength and direction of
$B_{\parallel}$ associated with molecular clouds111Tahani et al. (2018) code
for determining $B_{\parallel}$ is available at
https://github.com/MehrnooshTahani/MappingBLOS_MolecularClouds. In this
technique they incorporated an approach based on relative measurements to
estimate the amount of rotation measure (RM) caused by molecular clouds, using
RM data from Taylor et al. (2009). To extract the magnetic field strengths
from the RMs, they estimated the electron column density of the molecular
cloud at the position of each RM point using a chemical evolution code (Gibson
et al. 2009) and extinction maps for each cloud.

The observations of $B_{\perp}$ and $B_{\parallel}$ are slowly paving the way
for the determination of 3D magnetic fields of molecular clouds. Tahani et al.
(2019) investigated the 3D magnetic field morphology of the Orion A molecular
cloud using both $B_{\parallel}$ (Tahani et al. 2018) and $B_{\perp}$ (Planck
Collaboration et al. 2016c) observations. The line-of-sight magnetic field
observations in this region showed that the $B_{\parallel}$ reverses direction
from one side of this filamentary-shaped cloud to the other (perpendicular to
the filament axis; illustrated in Fig. 1). Tahani et al. (2019) constructed
models to account for this reversal and compared their synthetic observations
to the observed $B_{\perp}$ and $B_{\parallel}$. Using Monte-Carlo simulations
and chi-squared probability values (as well as examining systematic biases
between the two observing techniques), they concluded that an arc-
shaped222Also referred to as bow-shaped; pronounced /bō/ as in rainbow or bow
and arrow magnetic field morphology was the most likely magnetic structure for
the Orion A molecular cloud.

This arc-shaped morphology is illustrated in the inset of Fig. 1. The
background gray-scale image in this figure depicts the visual extinction map
of Orion A (in units of magnitude of visual extinction or $A_{{V}}$;
Kainulainen et al. 2009), and the red and drapery lines (made using the line
integration convolution technique333https://pypi.org/project/licpy/, Cabral &
Leedom 1993) show the $B_{\perp}$ observed by the Planck Space Observatory,
with blue (red) circles indicating magnetic fields toward (away from) us. The
magnetic field on the Galactic north side of the cloud points toward us, while
on the Galactic south side it points away from us. This $B_{\parallel}$
reversal was previously observed using Zeeman measurements in the study of
Heiles (1997), which predicted an arc-shaped magnetic field morphology caused
by recurrent shocks from nearby supernovae in the Orion-Eridanus superbubble.

Figure 1: The observed magnetic fields of the Orion A molecular cloud. The
$B_{\parallel}$ data were obtained by Tahani et al. (2018) and the $B_{\perp}$
lines were observed by the Planck Space Observatory. The blue (red) circles
show magnetic fields toward (away from) us, and the size of the circles
represents the strength of the magnetic fields. The red and drapery lines show
the $B_{\perp}$orientation. The background gray-scale image is the visual
extinction map obtained by Kainulainen et al. (2009). Tahani et al. (2019)
suggested that an arc-shaped magnetic field morphology (as illustrated in the
inset cartoon) is the most probable model that can explain the observed
$B_{\parallel}$ reversal across the cloud.

An arc-shaped magnetic field morphology has also been generated in the ideal
magnetohydrodynamic simulations of Inoue et al. (2018) and Inoue & Fukui
(2013), and predicted in the Inutsuka et al. (2015) molecular-cloud-formation
model (also see Inutsuka et al. 2016). Multiple compressions caused by
expanding ISM bubbles are required for the formation of filamentary molecular
clouds in the Inutsuka et al. (2015) model, where magnetic field lines with a
component perpendicular to the direction of bubble expansion or propagation
can be bent. Figure 2 depicts a schematic cartoon of this interaction, which
occurs between a relatively dense cloud ($\sim 10^{3}$ cm-3) and a shock wave
propagating in low density gas (H$I$1.2ex) and bends the field lines around
the formed filament. We refer to this cloud-formation model as the shock-cloud
interaction (SCI) model.

Figure 2: Formation of an arc-shaped magnetic field morphology around
filamentary molecular clouds as described by Inoue et al. (2018). After the
interaction between the cloud and shock-front, with the velocity $\@vec{v}$ in
the co-moving frame of the shock-front, a filamentary structure is formed
(shown in an end-on view in the right side). $\@vec{B}$ shows the direction of
the initial magnetic field before the collision and the morphology of the
magnetic field after the collision.

While previous studies described the magnetic field morphology of the Orion A
cloud as having an arc shape, its complete 3D orientation and direction
(particularly when projected onto the plane of the sky) remained unknown, even
in the 3D magnetic field study by Tahani et al. (2019). In this study, the
complete 3D magnetic field morphology of the Orion A molecular cloud,
including its direction and bending orientation, is reconstructed from data.
To this end, we employ Galactic magnetic field (GMF) models, the
$B_{\parallel}$ data of Tahani et al. (2018), and the $B_{\perp}$ observed by
the Planck Space Observatory (whereas in Tahani et al. (2019) only $B_{\perp}$
and $B_{\parallel}$ observations were used). We present our approach and
results in Sect. 2, discuss the role of surrounding structures, bubbles, and
events in formation of this 3D field in Sect. 3, and provide a summary and
conclusion in Sect. 4. Supplemental material, including the data used in this
study (see Appendix A), are provided in the appendix sections.

##  2 Results: 3D magnetic field morphology of Orion A

We recreate the 3D magnetic field shape of the Orion A molecular cloud using
GMF vectors as initial magnetic fields, the orientation of $B_{\parallel}$
reversal, and $B_{\perp}$ morphology (under the assumption of an arc-shaped
magnetic morphology). To approximate the direction of the initial magnetic
fields, we use the GMF model of Jansson & Farrar (2012), neglecting the
isotropic random field components (caused by ISM turbulence). We refer to this
structure as the “Coherent GMF” model. Figure 3 illustrates these GMF vectors
projected onto the plane of the sky (red arrows) at the location of Orion A.

To best describe the GMF vectors, we employ a frame of reference in this
region (see Fig. 4), with its axes pointing in the increasing directions of
longitude ($\hat{\ell}$), latitude ($\hat{b}$), and distance ($\hat{d}$) at
Orion A’s plane-of-sky location (where $\hat{\ell}$, $\hat{b}$, and $\hat{d}$
are unit vectors). The GMF direction in this region can be described as a unit
vector of $-0.7\hat{\ell}-0.1\hat{b}-0.7\hat{d}$. This vector appears mostly
parallel to Orion A when projected onto the plane of the sky, but has a large
component along the line of sight (pointing toward us). This vector is
consistent with previous GMF studies (e.g., Van Eck et al. 2011, see their
Fig. 6), with the same inclination angle and direction (45∘ to 50∘ with
respect to the plane of the sky; e.g., Oren & Wolfe 1995; Heiles 1997).

Figure 3: Galactic magnetic field associated with the Orion A cloud. The
background gray-scale image shows the extinction map from Kainulainen et al.
(2009). The yellow line represents the division between the tail and the head
of the cloud. The region of Orion A with longitudes less than $\sim
210^{\circ}$ is referred to as the “head”, while the portion with longitudes
greater than $\sim 210^{\circ}$ is referred to as the “tail” of Orion A. The
red vectors show the Coherent GMF model projected onto the plane of the sky.
The cyan line illustrates an approximate filament axis.

We also need to account for the inclination angle of Orion A when
reconstructing its 3D magnetic field. Großschedl et al. (2018) demonstrated
that the majority of Orion A has a high inclination angle with respect to the
plane of the sky. They divided the cloud into a head (longitudes less than
$\sim 210^{\circ}$) and a tail (longitudes greater than $\sim 210^{\circ}$, as
depicted in Fig. 3), with the head running approximately parallel to the plane
of the sky and the tail trending into the line of sight at an approximately
$70^{\circ}$ inclination angle, as illustrated in Fig. 4.

We reconstruct an arc-shaped morphology as depicted in Fig. 4, by connecting
the approximate GMF vectors to the $B_{\parallel}$ observations via an arc
that accounts for the 3D orientation of the cloud and matches the $B_{\perp}$
morphology and the relative $B_{\parallel}$ strengths. The initial magnetic
field (the GMF) is depicted in this figure as a red vector pointing toward us,
and the filament is shown with a gray cylindrical shape. The blue vector
illustrates the reconstructed arc-shaped magnetic field morphology, which
looks semi-convex (from our point of view) and points in the direction of
decreasing latitude.

This 3D magnetic field morphology is an approximate large-scale configuration
that ignores smaller-scale variations and fluctuations, enabling us to
determine the direction and orientation of the arc, as well as the direction
of magnetic fields projected onto the plane of sky. Therefore, small
variations of this field could be possible as long as they remain consistent
with the $B_{\parallel}$ and $B_{\perp}$ observations, as well as the GMF
vectors. While the majority of these variations are insignificant, rotations
up to maximum $50^{\circ}$ along the black arrow shown in the middle and lower
panels of Fig. 4 may be possible (also see Fig. 7), covering possible field
shapes from the front to the back side of the cloud (that are consistent with
the observations). The field shapes in this possible range remain semi-convex
from our point of view and point in the decreasing latitude direction when
projected onto the plane of the sky.

Moreover, while we did not directly consider field strengths when
reconstructing the field lines, the results are consistent with observed
strengths; the cloud has a stronger line-of-sight field component to its
Galactic north side than to its Galactic south side. This is consistent with
the $B_{\parallel}$ observations of Tahani et al. (2018) and Zeeman
measurements of Heiles (1997). The error-weighted average strength of the
$B_{\parallel}$ observations in both studies is twice as strong on the cloud’s
Galactic north side as on the cloud’s Galactic south side. Finally, while we
believe that this 3D field is the most probable and natural field morphology
for the Orion A cloud, it does not completely rule out other possibilities.
Future observations with a high rotation measure source density and improved
$B_{\perp}$ resolution will allow us to more accurately and precisely
determine the 3D fields.

Figure 4: Cartoon showing the 3D morphology of the Orion A magnetic field. The
bent gray cylindrical shape and the red vector depict the Orion A filamentary
structure and the mean 3D Galactic magnetic field in the region, respectively.
The blue arrow shows the 3D magnetic field morphology of the cloud and is
consistent with the $B_{\parallel}$ and $B_{\perp}$ observations shown in Fig.
1. Top panel: Projection of the cloud onto the plane of the sky. Middle and
lower panels: Different viewing angles of the cloud. The black arrow denotes
the direction in which rotations of up to maximum $50^{\circ}$ may be
possible.

##  3 Discussion

Our reconstructed 3D magnetic morphology is consistent with the SCI cloud-
formation model, in which the surrounding environment (e.g., bubbles, events,
structures, or sheet-like clouds) influenced and bent the field lines. In this
section we discuss the formation of this arc-shaped morphology and
environmental influences that shaped it.

###  3.1 Bubbles influencing the Orion A cloud

Orion-Eridanus superbubble: Orion A (located at $432\pm 2$ pc from the Sun;
Zucker et al. 2019) was likely shaped by energy release from the Orion OB
association (Bally 2008). From the Sun, the cloud is seen in the projected
interior of the Orion-Eridanus superbubble. Its distance puts it on the far
side of the superbubble, but its location within, along, or beyond the bubble
rim remains uncertain because of the complex composite structure of the
superbubble (Ochsendorf et al. 2015; Joubaud et al. 2019). The stellar content
(Voss et al. 2010), stellar age gradient (Bouy & Alves 2015; Zari et al.
2017), and X-ray temperature gradient (Joubaud et al. 2019) all indicate that
the bubble has likely evolved in time and space from a near (150-200 pc away)
to far distance (close to Orion A and B) after a 10- to 20-Myr-long series of
supernova events. The approximate outline of the superbubble (based on
Ochsendorf et al. 2015) is depicted in Fig. 5 as a white circle, with the
background light-green and orange colors representing thermal dust (at 545
GHz) and H$\alpha$ observations, respectively. Figure 6 illustrates a rough 3D
approximation of the location of this superbubble as a gray ellipsoid, based
on the 3D models of Pon et al. (2014) and Pon et al. (2016).

Großschedl et al. (2020b), Kounkel (2020), and Pellizza et al. (2005) found
coherent stellar proper motion associated with the Orion region and the
superbubble. Coherent proper motions observed by Großschedl et al. (2020a) and
Kounkel (2020) are associated with the young stellar objects in the Orion A
and B clouds, implying the influence of a feedback-driven event. Großschedl et
al. (2020a) refers to this event as Orion-BB (big blast), whereas Kounkel
(2020) links the coherent velocities to the Barnard’s loop. In general, these
coherent velocities are likely indicative of feedback-driven (e.g.,
supernovae) impacts on the Orion A and B clouds and their young stellar
objects.

Barnard’s loop: Barnard’s loop (Barnard 1894) is a complete bubble located
within the Orion-Eridanus superbubble (Ochsendorf et al. 2015). The Barnard’s
loop bubble, with an estimated age of $3\times 10^{5}$ yr, is expanding at a
velocity of 100 km s-1 (Ochsendorf et al. 2015), while the Orion-Eridanus
superbubble has an expansion velocity of $\sim 20$ km s-1 (Joubaud et al.
2019). This bubble is located between 340 pc and 400 pc from us (Großschedl et
al. 2020b). Using Barnard’s loop and the Orion A cloud’s distances, we find
that the Barnard’s loop bubble is likely in contact with (and has interacted
with) Orion A’s head but not its tail, as depicted in Fig. 6. This could
account for the tilt of Orion A’s head and the fact that the star formation
rate in Orion A’s head is an order of magnitude greater than in its tail
(Großschedl et al. 2020a). We predict that as the Barnard’s loop bubble
expands, it will interact with Orion A and trigger a new star-formation
sequence in the tail.

Orion dust ring: Schlafly et al. (2015) map the 3D dust and find a dust ring
(indicating a bubble origin) between 400 pc and 550 pc (from us) in the Orion
complex region. This dust ring is depicted in Fig. 5 as a green-dashed circle
and is seen at more positive latitudes compared to Orion A. Schlafly et al.
(2015) estimate the age of the bubble to be around 10 Myr or greater, because
they find no evidence of H$\alpha$ associated with the ring. A closer
examination of Fig. 5, however, indicates that this ring may be visible in
H$\alpha$ observations. We roughly approximate the ring (and its original
bubble) as a sphere (depicted in green in Fig. 6) with a radius of 150 pc and
a center location of $\ell=212^{\circ}$, $b=-11.5^{\circ}$, and $d=175$ pc.

Figure 5: Observations of the Orion-Eridanus superbubble and its nested
bubbles. The white, green dashed, and red dash-dotted circles illustrate the
approximate outline of the Orion-Eridanus superbubble, dust ring, and
Barnard’s loop on the plane of the sky, respectively. The light-green and
orange background colors represent the observations of thermal dust (at 545
GHz) obtained by the Planck Space Observatory and H$\alpha$ emission
(Finkbeiner 2003), respectively. The bright green region shows Orion A.
Figure 6: A simplified 3D approximation of the Orion-Eridanus superbubble and
its associated bubbles. Orion A is shown as an orange bent cylinder and the
Sun is depicted as a yellow circle. The gray ellipsoid and red and green
spheres represent the Orion-Eridanus superbubble, Barnard’s loop, and the dust
ring, respectively. Due to uncertainties in models of the Orion-Eridanus
superbubble, the dust ring might also be embedded within the superbubble.

###  3.2 Gradual formation of the arc-shaped field morphology

The bubbles detailed in Sect. 3.1 have influenced the evolution (and the
formation) of the Orion A cloud and its magnetic field lines. These influences
can be summarized in two steps: 1) recurrent interactions with old events or
bubbles pushed the Orion A cloud, its ambient environment, and the field lines
(toward more positive latitudes), resulting in large-scale bending of the
initial Galactic magnetic field, as predicted by the simulations of Kim &
Ostriker (2015). 2) Subsequently, further interactions (likely by the dust
ring) pushed the H$I$1.2ex gas around the Orion A cloud (toward more negative
latitudes), creating a sharp arc-shaped magnetic morphology associated with
the Orion A cloud, as depicted in Fig. 4.

This two-step process is similar to that described in Tahani et al. (2022) for
the Perseus molecular cloud (see Fig. 9 and Appendix D). In the first step,
the Galactic magnetic fields are bent approximately tangential to expanding
objects or bubbles (e.g., Orion-BB event or Orion-Eridanus superbubble). This
results in magnetic field lines that are bent at Orion A’s location, point
toward us on its Galactic north side, and are mostly parallel to the plane of
the sky on its Galactic south side (and perpendicular to the formed filament
main axis). Subsequently, the interaction with the dust ring bends these field
lines even more, causing them to point away from us on the cloud’s Galactic
south side. Therefore, the field lines on the Galactic north side of the cloud
have a greater component along the line of sight compared to the Galactic
south side of the cloud, as discussed in Sect. 2.

To examine this view further, we estimate the gas and magnetic field pressures
(see Appendix E) before the formation of the Orion A cloud and in its present
state. During the initial stages of the Orion A cloud formation, we find that
the gas pressure is greater than the magnetic field pressure (using a Galactic
magnetic field strength of $\sim 5\,\mu$G, particle volume density of a few to
100 cm-3, and ambient temperature of 1000 K and higher; Joubaud et al. 2019),
implying that recurrent supernovae can easily bend the magnetic field lines,
as seen in simulations of Kim & Ostriker (2015). Furthermore, using an average
non-thermal velocity dispersion of a few ($\sim 3$) km s-1, we find an Alfvén
Mach number of maximum $\sim 3$, indicating that the field lines can retain a
memory of the large-scale initial field morphology (with small-scale
variations; cf. Han & Zhang 2007). As a result, when the field lines interact
with bubbles and events in this region (such as the Orion-Eridanus superbubble
or the Orion-BB event) and bend, they remain largely coherent (Kim & Ostriker
2015), rather than becoming completely distorted and perturbed (Li & Klein
2019).

To estimate the gas and magnetic field pressures in the Orion A molecular
cloud in its current state, we use the error-weighted mean magnetic fields on
two sides of the cloud (for detections pointing toward us and away from us),
which are 87 $\mu$G and $-45$ $\mu$G, respectively, based on the observations
of Tahani et al. (2018). Using particle volume density and temperature values
of $\sim 10^{4}$ cm-3 (e.g., Castets et al. 1990; Dutrey et al. 1993;
Johnstone & Bally 1999a, b) and 25 K (e.g., Mitchell et al. 2001; Johnstone &
Bally 2006; Buckle et al. 2012; Schnee et al. 2014), we find that the magnetic
field pressure of the Orion A cloud is approximately one order of magnitude
greater than its gas pressure (on both sides of the cloud). Using an average
non-thermal velocity dispersion of 2 km s-1 (Goicoechea et al. 2020), we find
Alfvén Mach numbers of $0.4$ and $0.8$ for the two sides of the Orion A
molecular cloud. This implies that the field lines do not deform and if their
magnetic field morphology changes (due to interaction with the environment),
they retain a memory of their previous field morphology.

This gradual bending of the field lines is consistent with the cloud-formation
model of Inutsuka et al. (2015). We also determine the H$I$1.2ex and CO line-
of-sight velocities in this region (See Appendix C) and find no significant
offset between the two, indicating that the offset velocities caused by bubble
interactions have dissipated or are mostly in the plane of the sky. Given the
morphology of the Orion A cloud (which is inclined along the line of sight)
and the fact that the GMF vectors are parallel to the longitude axis and point
toward us along the line of sight, the interactions necessary for the
formation of the arc-shaped magnetic morphology are mostly parallel to the
plane of the sky. Therefore, we expect that any velocity offsets that may hint
at arc-shaped magnetic field morphology formation (and perhaps are not yet
completely dissipated) should be parallel to the plane of the sky.

##  4 Summary and conclusions

We determine the large-scale (and approximate) 3D magnetic field shape of the
Orion A molecular cloud using Galactic magnetic field models, as well as
present line-of-sight and plane-of-sky magnetic field observations. From our
perspective, this 3D field is generally semi-convex and points toward the
decreasing longitude and decreasing latitude directions in the plane of the
sky. To our knowledge, this is the first time that the complete 3D large-scale
($\sim$ a few to $\sim 100$ pc) magnetic field of the Orion A cloud has been
reconstructed (including its plane-of-sky direction). We suggest that the
Orion-Eridanus superbubble (or events within it, such as the Orion-BB event)
and the dust ring in this region are largely responsible for the development
of the arc-shaped magnetic field morphology of Orion A.

Reconstructing the 3D magnetic field morphology of Orion A, which is
consistent with the Planck observations when projected onto the plane of the
sky, relies mainly on the initial Galactic magnetic fields and the
$B_{\parallel}$ observations. This 3D magnetic field is a large-scale
approximation that neglects smaller-scale distortions or entanglements in the
field lines. Because we are interested in relatively large-scale and
approximate fields, the presence of low-density regions in the foreground of
Orion A (especially those smaller than the size of the Orion A cloud and the
scales at which the $B_{\parallel}$ reversal was observed; Rezaei Kh. et al.
2020) has no effect on our results.

###### Acknowledgements.

We would like to express our appreciation to the anonymous referee for their
thoughtful comments that helped us improve the results and paper. MT is
grateful for the helpful discussion with Pak-Shing Li. We have used LaTeX,
Python and its associated libraries including astropy (Astropy Collaboration
et al. 2013) and plotly (Plotly Technologies Inc. 2015), PyCharm, Jupyter
notebook, SAO Image DS9, the Starlink (Currie et al. 2014) software, the
Hammurabi code, Shapr3D, and Adobe Draw for this work. For our line
integration convolution plot, we used a Python function originally written by
Susan Clark. QuillBot444https://quillbot.com/ was used for language editing
purposes. The Dunlap Institute is funded through an endowment established by
the David Dunlap family and the University of Toronto. J.L.W. acknowledges the
support of the Natural Sciences and Engineering Research Council of Canada
(NSERC) through grant RGPIN-2015-05948, and of the Canada Research Chairs
program. Y.D. acknowledges the support of JSPS KAKENHI grant 18H01250. M.H.
acknowledges funding from the European Research Council (ERC) under the
European Union’s Horizon 2020 research and innovation programme (grant
agreement No 772663).

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##  Appendix A Data used in the study

In this study, we use estimates of the initial magnetic field direction, line-
of-sight magnetic fields, and velocities associated with each cloud. We use
the Jansson & Farrar (2012) GMF model to determine the Galactic magnetic
field. We employ the catalog of Tahani et al. (2018) for the $B_{\parallel}$
magnetic field information, and available CO and H$I$1.2ex observations for
the velocities.

###  A.1 Galactic magnetic field

The Jansson & Farrar (2012) model includes a two-dimensional (2D) thin-disk
field component that is tightly coupled to the Galactic spiral arms, an
azimuthal/toroidal halo field component, and an X-shaped vertical/out-of-plane
field component. To estimate the GMF, we use the Hammurabi
program555http://sourceforge.net/projects/hammurabicode/ (Waelkens et al.
2009), which is a synchrotron modelling code, that has been used in different
studies (e.g., Planck Collaboration et al. 2016b; Jansson & Farrar 2012). We
find the GMF vectors within a box around the Orion A cloud
($160\rm{\,pc}\times 160\rm{\,pc}\times 200\rm{\,pc}$), in the longitude range
of $205^{\circ}$ to $218^{\circ}$ and latitude range of $-26^{\circ}$ to
$-13^{\circ}$. We set a resolution of one GMF vector per $2\,\rm{pc\,}\times
2\,\rm{pc\,}\times 2\,\rm{pc}$.

###  A.2 Line-of-sight magnetic field

We employ the $B_{\parallel}$ observations of Tahani et al. (2018), in which
they used Faraday rotation measurements to determine $B_{\parallel}$ of
molecular clouds. They used RM point sources along with an on-off approach
based on relative measurements to decouple the molecular clouds’ contribution
to RM from that caused by the rest of the Galaxy. They then calculated the
strength of $B_{\parallel}$ employing a chemical evolution code and the
Kainulainen et al. (2009) extinction maps (for more details, see Tahani et al.
2018).

###  A.3 Velocity information

We consider both the available CO and H$I$1.2ex velocities to explore the
line-of-sight velocities of the cloud and its surrounding environment. We
obtain H$I$1.2ex velocity information from the all-sky database of the
H$I$1.2ex $4\pi$ Survey (HI4PI; HI4PI Collaboration et al. 2016), which is
based on the Effelsberg-Bonn H$I$1.2ex Survey (EBHIS; Kerp et al. 2011; Winkel
et al. 2016) and the Galactic All-Sky Survey (GASS; McClure-Griffiths et al.
2009). We use the radial velocities from the Dame et al. (2001) carbon
monoxide survey, to determine the cloud’s CO velocity. This catalog is a
survey of the 12CO J(1-0) spectral line of the Galaxy.

##  Appendix B 3D magnetic field sensitivity

As discussed in Sect. 2, the reconstructed 3D field is an approximation of the
large-scale field, which allows for small-scale variations such as small
rotations along the black arrow shown in the middle panel of Fig. 7. Rotations
in other directions, on the other hand, are more sensitive and limited, as
they do not produce the observed $B_{\parallel}$ or $B_{\perp}$. We emphasize
that these variations must remain consistent with the GMF, as well as the
$B_{\parallel}$ and $B_{\perp}$ observations. The magnetic field that we
reconstructed666The .obj files are available at
https://github.com/MehrnooshTahani/OrionA3DMagneticFields in Sect. 2 is
depicted as 3D B (1) in Fig. 7. Examining a large number of 3D field
varieties, we find that rotations along the black arrow up to maximum
$50^{\circ}$ may be possible (covering possible field morphologies from the
front to the back side of the cloud), resulting in the arc-shaped field
depicted as 3D B(2) in Fig. 7, which is semi-convex from our perspective and
points in the direction of decreasing latitude.

Figure 7: Cartoon showing the sensitivity of the reconstructed 3D magnetic
field. The Galactic magnetic field, the Orion A cloud, and the reconstructed
field are shown with the red vector, bent gray cylindrical shape, and the blue
3D B (1) arrow, respectively. Top panel: Projection onto the plane of the sky.
Middle and lower panels: Different viewing angles of the cloud. The black
arrow in the middle panel illustrates the direction of rotation to obtain 3D
B(2), which remains consistent with the observations and is semi-convex from
our point of view.

##  Appendix C Velocity observations

To investigate the line-of-sight velocities of the cloud thoroughly, we first
calculate the line-of-sight Galactic rotation velocities. We find a Galactic
rotation velocity (LSR) of $\sim+4$ km s-1 for the Orion A cloud ($l\simeq
211^{\circ}$ and $d\simeq 432$ pc), using the model of Clemens (1985) with the
IAU standard values of the solar distance from the Galactic center (8.5 kpc)
and orbital velocity (220 km s-1).

For H$I$1.2ex velocities, we select locations that have a single H$I$1.2ex
peak emission and exclude those with multiple peaks, absorption, or self-
absorption, as discussed in Tahani et al. (2022). Similarly, we pick points
whose CO spectrum can be described primarily by a single Gaussian fit. For
more accuracy, we take H$I$1.2ex and CO data from similar coordinates. These
H$I$1.2ex and CO velocities agree with other studies at similar coordinates
(e.g., Kong et al. 2015; Rice et al. 2016; Ishii et al. 2019; Ma et al. 2020).
We note that average CO and HI velocities along the main axis of the cloud are
close to Galactic rotation velocities at this location ($\sim 4$ to $6$ km
s-1).

Subsequently, we can find the molecular cloud velocities (CO) in the co-moving
frame of the H$I$1.2ex gas using:

$v_{\text{CO - H\scaleto{$I$}{1.2ex}}}=v_{\text{CO,
LSR}}-v_{\text{H\scaleto{$I$}{1.2ex}, LSR}},$ (1)

where $v_{\text{CO - H\scaleto{$I$}{1.2ex}}}$ is the cloud CO velocity in the
co-moving H$I$1.2ex frame, $v_{\text{CO, LSR}}$ is the CO velocity in the
local standard of rest (LSR) frame and $v_{\text{H\scaleto{$I$}{1.2ex}, LSR}}$
is the H$I$1.2ex velocity of the region in the LSR frame. To account for the
velocity gradients and fluctuations along the cloud, we find $v_{\text{CO -
H\scaleto{$I$}{1.2ex}}}$ on the cloud point by point (considering the peak
emission of CO and H$I$1.2ex) and then take an average. This obtained value is
$0.0\pm 0.5$ km s-1 (where the uncertainty value is the standard deviation of
the $v_{\text{CO - H\scaleto{$I$}{1.2ex}}}$ points), indicating that within
the uncertainty range of the observations, there is no significant line-of-
sight offset between the H$I$1.2ex and CO velocities in the Orion A cloud
(also consistent with Fig. 1 of Imara & Blitz 2011).

This zero km s-1 line-of-sight average CO velocity in the co-moving H$I$1.2ex
frame emphasizes the importance of considering the plane-of-sky environmental
influences and their effect on the GMF vectors. Furthermore, since the GMF
vectors fall parallel to the cloud on the plane of the sky and the cloud has a
large inclination angle, it is even more critical to consider plane-of-sky
influences and events (e.g., coherent velocities observed by Großschedl et al.
2020b) in order to fully understand the evolution of 3D magnetic fields in
this region.

##  Appendix D Orion bubbles

We discussed the bubbles that influence the Orion A cloud in Sect. 3.1 and
Sect. 3.2. The presence of these bubbles is further demonstrated by multi-
wavelength observations shown in Fig. 8. Thermal dust, H$\alpha$, CO, and
H$I$1.2ex observations are illustrated in this multi-panel figure, with the
Orion-Eridanus superbubble, dust ring, Barnard’s loop, and the $\lambda$
Orionis ring outlined as white, green dashed, red dash-dotted, and blue dotted
circles, respectively.

In Sect. 3.2, we discussed the gradual evolution of initial magnetic field
lines, resulting in the arc-shaped morphology depicted in Fig. 4. This
evolution process is summarized in Fig. 9: 1) The first step (left panel)
involves the bending of Galactic magnetic fields toward positive latitudes (by
the Orion-Eridanus superbubble, or bubbles and events within, such as the
Orion-BB event). At this stage field lines are bent at Orion A’s location,
pointing toward us on its Galactic north side, and mostly parallel to the
plane of the sky on the Galactic south side of the cloud (on the plane of the
sky appearing mostly perpendicular to the main axis of the filament that is
formed). 2) Following that (right panel), additional environmental influences,
such as the dust ring, interact with the cloud’s surrounding H$I$1.2ex gas,
further bending these field lines, causing them to point away from us on the
cloud’s Galactic south side.

Figure 8: Multi-wavelength observations of the Orion-Eridanus superbubble. The
Orion-Eridanus superbubble, dust ring, Barnard’s loop, and the $\lambda$
Orionis ring are depicted as white, green dashed, red dash-dotted, and blue
dotted circles, respectively. Top-left panel: Thermal dust observations from
the Planck Space Observatory. Top-middle panel: $H\alpha$ observations
(Finkbeiner 2003). Top-right panel: Composite CO survey of Dame et al. (2001).
Lower panel: H$I$1.2ex observations at velocities of $2.7$ km s-1 and $9.1$ km
s-1 from HI4PI.

Figure 9: Formation of the arc-shaped magnetic field morphology in the Orion A
cloud. The large light blue circle represents a bubble within the Orion-
Eridanus superbubble, or the Orion-Eridanus superbubble itself (progenitor(s)
represented by the star sign). The small dark blue cylinder illustrates the
Orion A filamentary molecular cloud. The 3D magnetic configurations cannot be
accurately represented in 2D and a more accurate representation of the field
lines is shown in Fig. 4. Left panel: The straight $\@vec{B}$ represents the
direction of the initial magnetic field prior to interaction with the
bubble/event, while the large-scale mild bending of the magnetic field
morphology after interaction with the bubble/event is represented by the
curved $\@vec{B}$. Right panel: Formation of the arc-shaped magnetic field
morphology around the Orion A molecular cloud due to further interactions,
possibly with the dust ring.

##  Appendix E Alfvén Mach number and gas and magnetic pressures

In Sect. 3.2, we calculated magnetic and gas pressures, as well as Alfvén Mach
numbers, to elucidate the step-by-step evolution of the field lines that
resulted in the arc-shaped morphology shown in Fig. 4. To estimate the
magnetic and gas pressures, we used the following equations,

$\displaystyle P_{B}=\frac{B^{2}}{8\pi}[cgs],$ (2) $\displaystyle
P_{\text{gas}}=nk_{b}T,$

where $P_{B}$, $B$, $P_{\text{gas}}$, $n$, $k_{B}$, and $T$ are the magnetic
pressure, total strength of magnetic field, gas pressure, particle volume
density, Boltzmann constant, and temperature, respectively.

To determine the Alfvén mach number, we used the following equation:

$\mathcal{M}_{A}=\frac{\sigma_{\varv}}{\varv_{A}},$ (3)

where $\sigma_{\varv}$ is the non-thermal velocity dispersion and $\varv_{A}$
is the Alfvén wave group velocity. The Alfvén velocity can be obtained using

$\varv_{A}=\frac{B}{\sqrt{4\pi\rho}}[cgs],$ (4)

where B is the strength of magnetic field and $\rho$ is the volume density.
Simulations by Li & Klein (2019) suggest that in the presence of strong fields
($\mathcal{M}_{A}\simeq 1$), magnetic field lines can coherently bend around a
formed filamentary molecular cloud as a result of converging flows, whereas
for $\mathcal{M}_{A}\simeq 10$, complete distortion of magnetic fields is
expected.