# Magnetic fields in star forming environments: how does field strength affect
gas on spiral arm and cloud scales?

Nicholas P. Herrington,1 Clare L. Dobbs,1 Thomas J. R. Bending1
1School of Physics and Astronomy, University of Exeter, Stocker Road, Exeter,
EX4 4QL, UK E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

We investigate star formation from subparsec to kpc scales with
magnetohydrodynamic (MHD) models of a cloud structure and a section of
galactic spiral arm. We aim to understand how magnetic fields affect star
formation, cloud formation and how feedback couples with magnetic fields on
scales of clouds and clumps. We find that magnetic fields overall suppress
star formation by $\sim$10% with a weak field (5 $\mu$G), and $\sim 50$% with
a stronger field (50 $\mu$G). Cluster masses are reduced by about 40% with a
strong field but show little change with a weak field. We find that clouds
tend to be aligned parallel to the field with a weak field, and become
perpendicularly aligned with a stronger field, whereas on clump scales the
alignment is more random. The magnetic fields and densities of clouds and
clumps in our models agree with the Zeeman measurements of the Crutcher
relation $B-\rho$ in the weaker field models, whilst the strongest field
models show a relation which is too flat compared to the observations. In all
our models, we find both subcritical and supercritical clouds and clumps are
present. We also find that if using a line of sight (1D) measure of the
magnetic field to determine the critical parameter, the magnetic field, and
thereby also criticality, can vary by a factor of 3-4 depending on whether the
direction the field is measured along corresponds to the direction of the
ordered component of the magnetic field.

###### keywords:

galaxies: star formation – ISM: clouds – methods: numerical – magneto - hydro-
dynamics – radiative transfer – HII regions

††pubyear: 2024††pagerange: Magnetic fields in star forming environments: how
does field strength affect gas on spiral arm and cloud scales?–C

##  1 Introduction

Magnetic fields permeate the interstellar medium (ISM) making them important
on all scales of the star formation process, from molecular clouds down to the
protostellar environment. Observations indicate that the median magnetic field
strength in the cold neutral medium (CNM) of the ISM is $\sim
6\leavevmode\nobreak\ \mu$G (Crutcher, 1999; Heiles & Troland, 2005; Crutcher
et al., 2010), but can be as high as $20-30\leavevmode\nobreak\ \mu$G in M51
and $50-100\leavevmode\nobreak\ \mu$G in M87 (see Beck 2015 and references
therein). At these field strengths, magnetic energy densities are comparable
to kinetic energy densities arising from turbulence, but smaller than thermal
energies except for the hottest phases of the ISM (Crutcher, 1999; Cox, 2005;
Beck, 2015). Thus magnetic fields could be expected to be as significant as
other processes in the ISM such as turbulence and feedback, and potentially
provide support against gravitational collapse on core or cloud scales.

The magnetic criticality of a cloud is a measure of whether a cloud will be
supported against collapse by magnetic fields or not. The magnetic criticality
can be computed by comparing a cloud’s mass to its critical mass
$M_{\scriptscriptstyle\textrm{crit}}$. The collapse of a cloud is expected to
proceed if its mass to flux ratio exceeds a critical value
$M/M_{\scriptscriptstyle\textrm{crit}}=\alpha_{B}\Phi_{\scriptscriptstyle
B}/G^{1/2}$, where $\Phi_{\scriptscriptstyle B}=B\pi R^{2}$ (Mouschovias &
Spitzer, 1976; Nakano, 1978; Krumholz, 2011) and $\alpha_{B}$ is $\sim$0.17
for a uniform spherical cloud. This can alternatively be expressed in terms of
observable quantities, e.g. $M/M_{\scriptscriptstyle\textrm{crit}}\sim
7.6\times 10^{-21}N(H_{\scriptscriptstyle
2})/B_{\scriptscriptstyle\textrm{TOT}}$, where $N$ is the column density and
$B_{\scriptscriptstyle\textrm{TOT}}$ is the magnetic field strength (Crutcher,
2004). A structure for which $M/M_{\scriptscriptstyle\textrm{crit}}>1$ is
referred to as supercritical. For mass to flux ratios below the critical value
(i.e. $M/M_{\scriptscriptstyle\textrm{crit}}<1$) structures are subcritical,
i.e. magnetically supported against collapse. We know that clouds are evolving
structures and so their criticality may not stay consistent throughout their
lifetimes. This has lead to two different pictures of magnetic fields and
their role in the star formation process, referred to as strong field and weak
field models (Shu et al., 1987; Crutcher, 2012). In strong field scenario
clouds are born subcritical, collapsing only once they become supercritical
through accretion of mass increasing their
$M_{\scriptscriptstyle\textrm{crit}}$ (Vázquez-Semadeni et al., 2011; Körtgen
et al., 2018) and or through ambipolar diffusion of the constituent gas and
ions (e.g., Parker 1967; Mouschovias 1979; Shu 1983).Weak field models (e.g.,
Padoan & Nordlund 1999; Mac Low & Klessen 2004) predict that clouds are
controlled by turbulent flows and either dissolve back into the interstellar
medium (ISM) or collapse if self gravitating upon formation. Vázquez-Semadeni
et al. (2011) looked at the role of turbulence and magnetic fields by
evaluating criticality and star formation properties of clouds structures
formed in converging flows. They find that the criticality of clouds is very
sensitive to location within the cloud, suggesting regions of a cloud can be
supercritical whilst globally magnetically supported. This was also found in
work from Hu et al. (2023), as well as concluding that most clouds are
supercritical and subcritical clouds are identified as such due to
observational biases.

Observations highlight the dependency of magnetic field strength with density
across scales from clouds to cores (see reviews by Crutcher 2012 and Pattle et
al. 2023). Line of sight Zeeman measurements suggest that molecular clouds are
mostly supercritical following a $B\propto n^{\kappa}$ relation where
$\kappa\approx 2/3$ (Crutcher et al., 2010). Theoretically, there are a number
of ways $B$ can be expected to depend on $n$ (Crutcher, 2012). The most
extreme cases are when gas is compressed along ($\kappa=0$ as the field does
not change) or perpendicular ($\kappa=1$) to the field. For the weak field
case, there is no preferred direction and $\kappa\approx 2/3$ (Mestel, 1966).
When the field is strong and ambipolar diffusion is acting to enable
compression, $\kappa\leq 0.5$ (Mouschovias, 1981; Mestel & Paris, 1984;
Mouschovias & Ciolek, 1999; Auddy et al., 2022). Observations also indicate a
transition from a flat relation at low densities, to a power law relation at
high densities, the transition appearing to occur at densities of $n\sim
100-1000$ cm-3. The transition is usually assumed to coincide with a change
from structures being subcritical to supercritical (Crutcher, 2012). Models of
a turbulent magnetised ISM by Auddy et al. (2022) including ambipolar
diffusion also find the transition for the $B-\rho$ occurs at a density
$\rho_{\scriptscriptstyle T}$ related to the initial Alfvén Mach number,
$\mathcal{M}_{\scriptscriptstyle\textrm{A0}}$.

Magnetic fields are likely to effect the structure of the interstellar medium
on different scales, and ultimately the star formation rate. Magnetic fields
provide an additional pressure, on large scales inhibiting the formation of
dense structures such as clouds and interarm spurs (Dobbs & Price, 2008).
Wibking & Krumholz (2023) find the addition of magnetic fields leads to a
factor of 1.5-2 decrease in the star formation rates on a galaxy scale. Other
studies which follow the growth of galactic magnetic fields identify a
reduction in the star formation rate once the magnetic field amplification
saturates in the cold phase (e.g. Wang & Abel 2009; Pakmor & Springel 2013;
Hennebelle & Iffrig 2014; Van Loo et al. 2015). On molecular cloud scales,
simulations similarly find reduced star formation rates (e.g. Price & Bate
2009; Padoan & Nordlund 2011; Peters et al. 2011; Federrath & Klessen 2012;
Federrath 2015; Körtgen & Banerjee 2015; Van Loo et al. 2015). At the level of
cores, the addition of magnetic fields is found to reduce fragmentation,
leading to the formation of fewer brown dwarfs and producing an Initial Mass
Function (IMF) in better agreement with observations (Price & Bate, 2008). An
example of an exception to this trend is a colliding flow study from Zamora-
Avilés et al. (2018). They find magnetic fields aligned to inflows result in
the formation of less turbulent cloud structures which more readily collapse
and form stars compared to purely hydrodynamic case.

The direction of the magnetic field in the ISM is a further observational
measure of the influence of the magnetic field on gas. Filamentary structures
within the diffuse ISM appear to be aligned either parallel (at lower column
densities) or perpendicular (at higher column densities) to their local
magnetic field (McClure-Griffiths et al., 2006; Planck Collaboration et al.,
2016; Soler et al., 2017; Alina et al., 2019; Lee et al., 2021). Simulations
often find the transition between parallel and perpendicular alignments occurs
with increasing magnetic field strength and/or increasing density (e.g. Soler
et al. 2013; Klassen et al. 2017; Xu et al. 2019; Seifried et al. 2020; Dobbs
& Wurster 2021; Barreto-Mota et al. 2021; Mazzei et al. 2023). Girichidis
(2021) additionally identifies this transition and finds that it coincides
with a change in the angle between the flow and the magnetic field (i.e.
parallel or perpendicular). The transition from parallel to perpendicular
orientations has been related to shocks from colliding streams of gas (Inoue &
Inutsuka, 2009, 2016; Dobbs & Wurster, 2021), strongly converging flows in
turbulent clouds (Soler & Hennebelle, 2017), and the transition from
magnetically supported to gravitationally dominated filaments (Soler &
Hennebelle, 2017; Girichidis, 2021).

Most studies to date focus on either molecular cloud or whole galaxy scales.
In this study we investigate the evolution of a kpc scale star forming region,
consisting of a section of galactic spiral arm extracted from a pre-evolved
galaxy model. We additionally investigate a 100 pc scale cloud structure,
similarly extracted from a pre-evolved galaxy model. These models include
magnetohydrodynamics (MHD) and stellar feedback. We aim to understand the
magnetic criticality of cloud and clump structures, the orientation of
filaments with respect to elongation to their magnetic fields and how magnetic
fields interact with stellar feedback. This paper is structured in the
following way: in Section 2 we describe our simulation setup, in Section 3 we
show the evolution of our Arm and Cloud scale models, the magnetic properties
of cloud and clump structures, the magnetic field density relation, the
interaction between feedback and magnetic fields and lastly the properties of
cluster in our pc scale models. In Section 4 we discuss methods to calculate
the critical mass to flux ratio, a discussion of magnetic fields from the
context of our models. A summary of our main results is given in Section 5.

##  2 Methodology

Our simulations are modelled using the magnetohydrodynamics code sphNG (Benz,
1990; Benz et al., 1990; Bate et al., 1995a; Price & Monaghan, 2007),
featuring ISM chemistry for heating and cooling (Glover & Mac Low, 2007),
magnetic fields (Price, 2012) and non ideal MHD (Wurster et al., 2014, 2016)),
sink particles (Bate et al., 1995b) and self gravity. Shocks are captured via
artificial viscosity (Morris & Monaghan, 1997), where the viscosity parameter
can vary between 0.1 and 1. The default smooth particle magnetohydrodynamics
parameters for artificial resistivity are used, i.e.
$\alpha_{\scriptscriptstyle B}=1$. The code is parallelised with both OpenMP
and message passing interface (MPI).

###  2.1 Magnetohydrodynamics

The simulations in this paper all assume ideal MHD. To keep the magnetic
divergence close to zero, we use a hyperbolic divergence cleaning method
(Dedner et al., 2002). This method couples the magnetic field to a scalar
field, through which the divergence error can be subtracted by propagating
$\nabla\cdot\textbf{B}$ as a damped wave. We use fast MHD waves with wave
speed $c_{\scriptscriptstyle h}$ for the divergence cleaning, which is
sufficient to enforce $h\nabla\cdot\textbf{B}/|\textbf{B}|<0.01$. The cleaning
is controlled by an over-cleaning parameter which we set as $\sigma=1$, the
default value. The magnetic field evolution follows the quantity
$\textbf{B}/\rho$, with the magnetic tension instability handled by the Børve
et al. (2001) method, see Price (2012) for further details.

Table 1: List of models, their initial magnetic field strengths, simulation
lifetime, progenitor galaxy model and mass resolution. G1 and G2 refer to the
galaxy models from Dobbs & Pringle (2013) and Pettitt et al. (2015)
respectively.

Model | Initial magnetic | Simulation | Galaxy | Mass resolution
---|---|---|---|---
Name | field strength | time (Myrs) | progenitor | (M⊙)
| ($\mu G$) |  | (G1/G2) |
Arm1 | 1 | 5.6 |  |
Arm5 | 5 | 6.8 | G1 | 3.67
Arm10 | 10 | 5.5 |  |
Arm0 | - | 5.8 |  |
Cloud5 | 5 | 6.3 |  |
Cloud50 | 50 | 7.1 | G2 | 0.02
Cloud0 | - | 6.0 |  |

Figure 1: Column density snapshots of Cloud0, Cloud5, Cloud50 at times of 1, 3
and 5 Myrs. Clumps are overlayed in colour based on their morphology. Green
clumps are filamentary, blue clumps are sheet like, turquoise are curved
sheets and red clumps are spheroidal. Sink particles are plotted in white.
Figure 2: Column density snapshots of models Arm0, Arm1, Arm5 and Arm10 at 6
Myrs. White points are cluster sink particles, green are filaments, blue are
sheets, red spheroidal and turquoise are curved sheets.

###  2.2 Simulation details

The simulations performed here largely use the same setup as those presented
in Herrington et al. (2023) and Dobbs et al. (2022a), but now focus on
magnetic fields. Our simulations based on Herrington et al. (2023) follow a
kpc size region and contain an initial population of stars. The initial
conditions are taken from a larger scale galaxy simulation (see Section 2.3),
including sink particles which contain a population of stars. Stars are
assigned to these sinks, using a method described later in this section. We
refer to these as ‘Arm’ models. Simulations based on Dobbs et al. (2022a)
follow a smaller, 100 pc region without an initial population and are referred
to as ‘Cloud’ models. In all the models, we also insert sink particles when
gas exceeds given densities, following Bate et al. (1995c). The sinks
represent small clusters of stars and so can also be thought of as cluster
sinks. Cluster sink particles are inserted when gas exceeds densities of 3000
cm-3 and 1000 cm-3 in the Arm and Cloud models respectively, and can accrete
gas with accretion radii of 0.6 and 0.1 pc.

Stellar feedback from massive stars is modelled by a prescription for
photoionising and supernovae feedback. For photoionisation a line of sight
(LOS) approach is used as described in Bending et al. (2020), in which LOS are
drawn for each gas particle around an ionising source. Along each LOS the
column density contribution of intersecting particles is added to the total
column density. The on-the-spot approximation is used for emission of 13.6 eV
photons through recombination, with a recombination rate of $2.7\times
10^{-13}$ cm3s-1. Photoionised gas is heated to $10^{4}$ K and can begin
cooling once its ionised fraction drops below 1%. Supernovae (SNe) are
inserted using the same method as Dobbs et al. (2011). The ambient gas density
is calculated from the neighbouring gas particles, which is then used to
calculate the velocity and temperature to be assigned to the gas, assuming one
supernovae deposits $10^{51}$ ergs of energy. The injected SNe represent the
pressure-driven snowplough phase, since the prior free expansion and adiabatic
phases are not necessarily well resolved.

As in previous work, we use a star formation efficiency (SFE) parameter
referring to the fraction of a cluster sink’s mass that we assume to be star
mass. This parameter is 0.25 in the Arm models, and 0.5 in the Cloud models.
We allocate stars to the sink particles using a pre-computed sample of stars
(Bending et al., 2020). They are inserted into sinks throughout the evolution
of both models as well as into the initial population of sinks in the Arm
models. The pre-computed sample has an associated massive star injection
interval, $\Delta M_{i}$, of 305 M⊙ for this work. This value is calculated by
dividing the total stellar mass of the sample by the number of stars above 8
M⊙. Massive star insertion occurs each time the total sink mass multiplied by
the SFE parameter increases by $\Delta M_{i}$. For each cluster sink particle
we find the target star mass, defined as the particle mass multiplied by the
SFE parameter. We also find the current star mass, defined as the number of
massive stars already added to the sink multiplied by $\Delta M_{i}$. We then
subtract the current mass from the target mass for each cluster sink particle.
If the sum of these differences is greater than $\Delta M_{i}$ we add another
massive star to the cluster sink particle with the maximum difference. The
lifetimes and stellar fluxes of the massive stars are calculated using the
stellar evolution program SEBA (Portegies Zwart & Verbunt, 2012). Only stars
with masses > 18 M⊙ are considered to be ionising, as the ionising output from
the lower mass end of the sample is negligible. Supernovae occur for all stars
> 8 M⊙ once they have reached their lifetime within the simulation.

###  2.3 Initial conditions

We present models on two differing scales, which as discussed in the previous
section we split into ‘Arm’ and ‘Cloud’ models. Arm models are split into four
simulations, three MHD and one HD model. Cloud models are split into three,
with two MHD and one HD model. The initial magnetic field strengths of each
MHD model are shown in Table 1. We label our models based on their initial
magnetic field strength, i.e. $B_{0}=0,1,5$ and $10\leavevmode\nobreak\ \mu$G
corresponds to Arm0, Arm1, Arm5 and Arm10. Our Cloud models follow the same
naming convention. The initial conditions for both the Arm and Cloud models
are extracted from pre-evolved galaxy models.

The initial conditions of our Arm models are setup as a combination of IOSFB
and MHD5 from Herrington et al. (2023), in which we extracted a 1.2 kpc3 box
from a galaxy evolution model from Dobbs et al. (2017) and apply a clockwise
toroidal magnetic field across the region. The region is in the upper left
quadrant of the galaxy, the initial magnetic field geometry approximately
follows the spiral arm and so is pointing predominantly along the y-axis. The
galaxy model included ISM heating and cooling, self gravity and a prescription
for supernovae feedback. Once we extract the gas from the galaxy we perform a
resolution increase of the same method as Bending et al. (2020), to improve
our mass per particle resolution from 312.5 M⊙ to 3.68 M⊙. We additionally ran
a couple of lower resolution simulations of the Arm5 model, to test the effect
of resolution, and the modelling of turbulence on the smallest scales in our
simulations. We discuss these simulations in Section 3.2.3, Section 5 and
Appendix A. As well as this, we discuss the impact of our initially uniform
magnetic field distribution in Section 5 and Appendix C.

The second set of initial conditions for our Cloud models are based on the
M1R1FB simulation from Dobbs et al. (2022b), which models a region of around
100 pc size scale. The initial conditions were similarly extracted from a
galaxy scale simulation from Pettitt et al. (2015) using the same resolution
increase scheme. Again a toriodal magnetic field is applied, with initial
field strengths of 5 and 50 $\mu$G, as the region is in the upper right
quadrant of the galaxy the initial magnetic field geometry points in a south-
eastern direction.

Similarly to Wurster et al. (2021) we include a surrounding isothermal
boundary which represents a warm ionised medium. Particles are placed within a
volume around the region such that they are in pressure equilibrium with the
extracted region. These boundary particles are fixed to a temperature of
$2\times 10^{4}$ K. We set the velocity of a boundary particle to match that
of the closest particle from the extracted region. This boundary medium is
included in both the Arm and Cloud scale models.

For both the Arm and Cloud models, the simulations differ from previous
simulations due both to the addition of the magnetic field, and the boundary
conditions. Because the addition of the boundary conditions mean the initial
conditions are not the same even without magnetic fields, non-MHD calculations
were also carried out for comparison.

Figure 3: The top panel shows the ratio of gas pressure to magnetic pressure,
plasma $\beta$, in which the dashed line indicates
$P_{\scriptscriptstyle\textrm{gas}}=P_{\scriptscriptstyle\textrm{mag}}$. The
lower panel shows the star formation rate surface density for hydrodynamical
and MHD models. Hydrodynamical models are shown as grey lines.

##  3 Results

###  3.1 Evolution of Arm and Cloud models

Figure 4: Histograms of the angle between a filament’s long axis and its
internal magnetic field, $\theta_{\scriptscriptstyle\textrm{rB}}$, for all Arm
and Cloud MHD models. These angles are calculated in 2D, including three
slices: 2D${}_{\scriptscriptstyle\textrm{xy}}$,
2D${}_{\scriptscriptstyle\textrm{xz}}$ and
2D${}_{\scriptscriptstyle\textrm{yz}}$, computed for each model.
$\theta_{\scriptscriptstyle rB}$ can range between 0∘ and 180∘, with angles
close to 0∘ indicating parallel and close to 180∘ being anti-parallel.

We show the evolution of the cloud scale models Cloud0, Cloud5 and Cloud50, at
times of 1, 3 and 5 Myr, in Figure 1. Comparing the control model (Cloud0) to
the MHD runs Cloud5 and Cloud50 we see that increasing the field strength
leads to much smoother structures. At 3 Myr, Cloud0 and Cloud5 are similar in
structure, displaying feedback bubbles produced by ionising massive stars. In
Cloud50 however these same feedback bubbles are not apparent, and the gas
remains relatively smooth and undisturbed. By 5 Myr both Cloud0 and Cloud5
show significant gas dispersal from the ionising massive stars in the clouds.
Within Cloud50 we start to see bubbles swept out by feedback, but there is
comparatively less gas dispersal. In all models we observe a large central
cluster formed from the collapse of the main filamentary structures threading
the middle of the clouds. The population of stars within Cloud0 and Cloud5
form around 3 further small but distinct clusters after 5 Myrs, but we only
clearly see the one central large cluster in Cloud50.

Figure 5: The mass to flux ratio, $\mu_{\scriptscriptstyle B}$, is shown in
the top row and the ratio of the Alfvén speed to the sound speed,
$\nu_{\scriptscriptstyle A}/\nu_{\scriptscriptstyle s}$, is shown in the
bottom row. These are calculated for clumps in Cloud5 and Cloud50 models at
times 1, 3 and 5 Myrs. The initial mass to flux ratios of Cloud5 and Cloud50
are also plotted, marked as a cross. The green contours correspond to the
diamond points at t = 1 Myrs. Circle points are associated with the orange
contours at t = 3 Myrs. Triangle points represented by the blue contours refer
to times t = 5 Myrs. The contours are created using a Gaussian Kernel Density
Estimate (KDE) function where the bandwith is calculated using Scott’s rule.

We show column density snapshots of our arm regions Arm0, Arm1, Arm5 and Arm10
at 6 Myr in Figure 2. Similarly to the cloud models, we see the gas structure
are increasingly smoothed with increasing initial magnetic field strength in
the arm models. Feedback bubbles are smoother and less structured in the
higher field strength models. Filamentary and sheet like clouds that form
around feedback bubbles are compressed and fragmented in the magnetic field
models, with increasing fragmentation occurring with increasing field
strength. In our non magnetic model Arm0 we see the largest number of sink
particles and cloud structures across the spiral and inter arm regions. With
increasing field strength, we see fewer sinks and cloud structures,
particularly in the inter-arm regions.

In Figure 3, we show the evolution of the ratio of the gas pressure to
magnetic pressure (plasma $\beta$) and the star formation rate surface density
($\Sigma_{\scriptscriptstyle\textrm{SFR}}$) for all models. Initially, the
plasma $\beta$ parameter (second panel) for Cloud5 decreases, indicating that
Cloud5 becomes more magnetically dominant, but this is then reversed and
Cloud5 becomes more gas pressure dominant after a few million years. This is
because the density increases as collapse occurs within the cloud, and also as
feedback occurs the temperature increases. Similar, fluctuating behaviour is
seen for Arm1, but for the higher strength models, where the magnetic pressure
is higher, there is less change in the densities compared to the initial
conditions. Cloud50 remains dominated by magnetic pressure ($\beta\lesssim
0.1$) during its evolution, and likewise magnetic pressure dominates for the
Arm5 and Arm10 models. In all these magnetically dominated models there is
little variation in $\beta$.

The amount of star formation is larger in Cloud5 than Cloud50, with the star
formation rate roughly double at all times. We expect the reduced star
formation in Cloud50 results in less feedback and the reduced level of gas
dispersal that we see in Figure 1. The Arm models follow a similar trend to
the Cloud models, i.e. a stronger initial magnetic field strength results in
lower star formation rates. Similarly to models in Herrington et al. (2023)
the initial star formation rates in our new arm models are roughly equal
during the first 2 million years. After this the magnetic models Arm5 and
Arm10 begin to plateau at lower star formation rates compared to Arm0. Arm0
and Arm1 exhibit similar star formation rates up to 4 Myr before the star
formation rate is suppressed in Arm1.

###  3.2 Clump and Cloud properties

####  3.2.1 Morphologies

Figure 6: The mass to flux ratio, $\mu_{\scriptscriptstyle B}$, is plotted in the top row and the ratio of the Alfvén speed to the sound speed, $\nu_{\scriptscriptstyle A}/\nu_{\scriptscriptstyle s}$, in the bottom row. These are calculated for clouds in Arm1, Arm5 and Arm10 models at times 2, 4 and 6 Myrs. The green contours correspond to the diamond points at t = 2 Myrs. Circle points are associated with the orange contours at t = 4 Myrs. Triangle points represented by the blue contours refer to times t = 6 Myrs. The contours are created using Gaussian KDE function where the bandwith is calculated using Scott’s rule. Table 2: Percentage of clouds that are classified as sheet, curved sheet, filament and spheroidal for Arm and Cloud models. These are calculated at times of 6 Myrs and 5 Myrs for the Arm and Cloud models respectively. Model | Sheet (%) | Curved sheet (%) | Filament (%) | Spheroidal (%)
---|---|---|---|---
Arm0 | 44.13 | 1.635 | 52.79 | 1.442
Arm1 | 42.32 | 1.289 | 53.71 | 2.685
Arm5 | 40.29 | 2.408 | 55.38 | 1.926
Arm10 | 41.20 | 1.547 | 54.35 | 2.901
Cloud0 | 43.12 | 0.2200 | 52.48 | 4.180
Cloud5 | 40.12 | 0.4559 | 55.17 | 4.255
Cloud50 | 33.40 | 0.2024 | 64.17 | 2.227

In order to identify clumps or clouds within the Cloud and Arm models, we use
a friends of friends algorithm (see Bending et al. 2020; Ali et al. 2022;
Herrington et al. 2023). The algorithm uses a particle separation to compare
the distance between a chosen particle and a potential group member. If this
distance is within the defined particle separation the particle is added to
the group. This process is repeated for ungrouped particles until all
particles in the model are checked. The particle separation is 0.8 pc for the
Arm models and 0.02 pc for the Cloud models. These length scales impose a
minimum density for identified clouds and clumps, which is equal to
$n_{\scriptscriptstyle\textrm{H}}\sim 200$ cm-3 and
$n_{\scriptscriptstyle\textrm{H}}\sim 2200$ cm-3 for the clouds (Arm models)
and clumps (Cloud models) respectively. With clumps and clouds identified, we
classify their shapes using the principle axes theorem. We solve the
eigenvalue problem for the moment of inertia equation
($\mathbf{L}=\mathbf{I\omega}$) to obtain the principle axes of inertia for a
given clump or cloud. This provides us with eigenvalues and vectors for the
corresponding axes which we use to classify the cloud’s / clump’s morphology.

In the same way as Ganguly et al. (2023) we can sort the eigenvalues for the
principle axes into $a,b$ and $c$, where $a>b>c$, and define four morphologies
as follows

$\textrm{{Filament:}}\ \frac{a}{b}>f_{\textrm{ar}};$ $\textrm{{Sheet:}}\
\frac{a}{b}\leq f_{\textrm{ar}}\ \&\ \frac{a}{c}>f_{\textrm{ar}};$
$\textrm{{Curved sheet:} \ {sheet but the centre of mass is outside the
object}};$ $\textrm{{Spheroidal:}}\ \frac{a}{c}\leq f_{\textrm{ar}}.$

$f_{\textrm{ar}}$ is the aspect ratio factor which is set to 3. An example of
each classification is shown in Figure 11. The clumps and clouds are also
shown in Figure 1, coloured according to morphology. In Table 2, we break down
the cloud or clump classifications as a percentage of the total number of
structures in each model. The most frequent morphology is filamentary,
followed by sheet-like, followed by spheroidal. Over time the percentage of
sheets structures increases in all Cloud models. Cloud0 shows a roughly equal
split between filaments and sheets, Cloud5 is more weighted towards filaments
and Cloud50 shows significantly more filaments then sheets, thus a stronger
magnetic field results in a higher fraction of filaments. This is similar to
results from Ganguly et al. (2023), as they also report a split in sheet like
and filamentary structures along with a very small percentage of spheroidal
structures. The Arm models similarly show clouds evenly distributed between
sheets and filaments, and again some tendency for filaments to become slightly
more common with increasing field strength. We also see that the number of
curved sheets and spheroidals encompasses only around $\sim 3\%$ of structures
in both the Arm and Cloud models.

####  3.2.2 Orientation of magnetic field and filaments

Figure 4 shows the most common angle,
$\theta_{\scriptscriptstyle\textrm{rB}}$, between the long axis of a filament
$r$ and its magnetic field $B$, for all Cloud and Arm models at times 5 and 6
Myrs respectively. We have taken three different 2D slices,
2D${}_{\scriptscriptstyle\textrm{xy}}$, 2D${}_{\scriptscriptstyle\textrm{xz}}$
and 2D${}_{\scriptscriptstyle\textrm{yz}}$, which are analogues to a bird’s
eye view of the mid-plane (xy) and looking through the mid-plane (xz and yz)
111The distribution of angles between two random vectors in 3D is biased
towards perpendicular, so even if the magnetic field has no relation to the
cloud axis it will appear preferentially perpendicular (Soler & Hennebelle,
2017). In the lowest magnetic field model, Arm1, we observe that clouds are
generally orientated parallel to the magnetic field, with the majority of
angles $\theta_{\scriptscriptstyle\textrm{rB}}$ between 0∘ to 30∘ (parallel)
and 160∘ to 180∘ (anti-parallel). This is true in all three 2D slices,
indicating the same trend regardless of viewing position. With increasing
field strength, we see relatively fewer parallel orientations. Arm5 shows no
preferred alignment of the clouds and their magnetic fields across all three
2D slices. Arm10 however does contain more perpendicularly aligned clouds (in
all three 2D slices) suggesting that perpendicular alignments are favoured in
higher magnetic field strength clouds. There is less of a trend for the Cloud
models, although there are still relatively more parallel aligned clouds in
Cloud5, and more clouds which are perpendicularly or closer to perpendicularly
aligned in Cloud50.

Other numerical studies find a transition from filaments tending to be aligned
parallel to the magnetic field, to being aligned perpendicular to the magnetic
field, at high densities and with sufficiently high magnetic field strengths
(Soler & Hennebelle, 2017; Seifried et al., 2020; Dobbs & Wurster, 2021;
Mazzei et al., 2023). There is also observational evidence of a transition
with density (Planck Collaboration et al., 2016; Soler et al., 2017; Alina et
al., 2019; Lee et al., 2021), though the distribution can be more random for
high contrast filaments (Alina et al., 2019) or on small, core scales (Chen et
al. 2020; Sharma et al. 2022). We see the Arm models agree with a dependence
on the magnetic field. A further test, Arm50 (with $B=50\leavevmode\nobreak\
\mu$G) showed filaments preferentially perpendicular to both the field and the
spiral arm compared to the other models. Both the Arm and Cloud models are
quite complex (compared to simple collisional cloud models such as Dobbs &
Wurster 2021) and as such, we might expect that locally filaments occur in
environments with different densities and field strengths, leading to both
parallel and perpendicular orientations, diluting the tendency for a clear
preferred orientation.

####  3.2.3 Magnetic criticality and magnetic energy

We display the magnetic critical parameter $\mu_{\scriptscriptstyle B}$, which
is the mass to flux ratio of a clump or cloud (subscript c refers to clump or
cloud),

$\mu_{\scriptscriptstyle
B}=\frac{M_{\scriptscriptstyle\textrm{c}}}{M_{\scriptscriptstyle\Phi}},\ \ \ \
\ \text{where}\ \ \ \ \
M_{\scriptscriptstyle\Phi}\equiv\frac{0.12}{\sqrt{G}}\Phi_{\scriptscriptstyle
B},\ \ \ \ \ \text{and}\ \ \ \ \ \Phi_{B}=B\pi
R^{2}_{\scriptscriptstyle\textrm{mag}},$ (1)

in the top panel of Figure 5 for clumps, and Figure 6 for clouds. The magnetic
field, $B$, is calculated from the average magnetic field strength of
particles in a structure. $R_{\scriptscriptstyle\textrm{mag}}$ is the radius
used to work out the flux, $\Phi_{\scriptscriptstyle B}$, through the centre
of mass of the clump or cloud. This radius is calculated as

$R_{\scriptscriptstyle\textrm{mag}}=\frac{R_{\scriptscriptstyle\textrm{COM}}}{3}\left(1+\frac{b}{a}+\frac{c}{a}\right).$
(2)

where $R_{\scriptscriptstyle\textrm{COM}}$ is the radius of the clump
calculated as the furthest particle away from the centre of mass (COM). $a$,
$b$ and $c$ are the eigenvalues of the principle axes calculation mentioned
early in this section. This equation takes into account the morphology of the
clump, and avoids overestimating the flux surface for clumps that are not
spherical, i.e. filaments and sheets. The mass to flux ratio,
$\mu_{\scriptscriptstyle B}$, defines the magnetic criticality of a structure.
For subcritical clouds, $\mu_{\scriptscriptstyle B}>1$, and the structure is
magnetically supported whilst for supercritical clouds,
$\mu_{\scriptscriptstyle B}<1$, and magnetic fields cannot support the
structure against collapse.

For the cloud models (Figure 5), Cloud5 is initially globally supercritical
and Cloud50 globally subcritical. For clumps within the cloud though there is
a spread in criticality, which increases with time. At early times, most
clumps are subcritical in both models, but at later times there are also many
supercritical clumps, and Cloud50 unexpectedly contains some strongly
supercritical clouds. The strongly supercritical clumps we see in Cloud50 at 5
Myr are all small, dense, low mass clumps, which result from fragmentation and
compression as feedback acts on the gas. In Cloud5 and Cloud0, the feedback
simply disperses the gas and fewer such clumps survive. We also see that there
is a much larger spread in $\mu_{\scriptscriptstyle B}$ for low mass clumps.
Higher mass ($10^{3}-10^{4}$ M⊙) clumps tend to have $\mu_{\scriptscriptstyle
B}$ close to 1.

For the Arm models (Figure 6), most of the clouds are supercritical. The
higher magnetic field strength models have more subcritical clouds as would be
expected. There is some increase in spread in $\mu_{\scriptscriptstyle B}$
with time, but less than the Cloud models, and again the more massive clouds
have the lowest degree of spread. The most supercritical clouds are similarly
small, low mass structures. The development of small dense supercritical
clumps is less evident in the Arm models, presumably because they are not
resolved on these scales or the field strengths are too weak.

In the lower panels of Figure 5 and 6 we show the ratio of the Alfvén speed to
the sound speed of the gas, i.e. $\nu_{\scriptscriptstyle
A}/\nu_{\scriptscriptstyle s}$, where $\nu_{\scriptscriptstyle
A}=B_{\scriptscriptstyle\textrm{c}}/(4\pi\rho_{\scriptscriptstyle\textrm{c}})^{\scriptscriptstyle
0.5}$ and $\nu_{\scriptscriptstyle s}=((\gamma^{\scriptscriptstyle
2}-\gamma)u_{\scriptscriptstyle\textrm{c}})^{\scriptscriptstyle 0.5}$, where
$\gamma=2.381$, $u_{\textrm{c}}$ is the internal energy, subscript c is cloud
or clump. This ratio provides an indication of the dominant information speed
within a structure. $\nu_{\scriptscriptstyle A}/\nu_{\scriptscriptstyle s}>1$
indicates magnetic information propagates faster than the gas information,
suggesting a structure which is magnetically dominated. In Cloud5 and Cloud50
the structures are mostly supercritical and so correspondingly the dominating
information speed is the sound speed. Again, the compression by feedback
increases the density and temperature of the clumps, which decreases
$\nu_{\scriptscriptstyle A}$ and increases $\nu_{\scriptscriptstyle s}$. These
kinematically driven clumps become increasingly supercritical since the
magnetic field is much less responsive. The Arm models similarly show that
most clumps are dominated by the sound speed. The ratio
$\nu_{\scriptscriptstyle A}/\nu_{\scriptscriptstyle s}$ shows a similar trend
to $\mu_{\scriptscriptstyle B}$, i.e. with increasing field strength,
$\mu_{\scriptscriptstyle B}$ decreases and correspondingly
$\nu_{\scriptscriptstyle A}/\nu_{\scriptscriptstyle s}$ increases.

We also computed the ratios of magnetic energy to the kinetic and
gravitational energies. The ratios of the energies largely agree with the
critical parameter, and most clouds have higher kinetic and gravitational
energies compared to the magnetic energy. Other numerical work also finds that
clouds tend to have greater gravitational energies (Hu et al. 2023; Ganguly et
al. 2023), though Hu find the critical parameters can be subcritical when
calculated using synthetic observations. Ganguly et al. (2024) also find that
the magnetic surface term can be a significant confining pressure,
particularly for lower density clouds.

In Appendix A we test whether resolution changes the magnetic properties of
our models, and compare the results from model Arm5 with a lower resolution
version of Arm5. We do not see a significant difference between the magnetic
properties of the clouds at lower resolution, indicating that our conclusions
about magnetic fields are not strongly resolution dependent.

###  3.3 Magnetic field density relation

The top Panel of Figure 7 shows the magnetic field density relation $B-\rho$
for our Cloud and Arm models (in the right and left panels respectively). We
compare this to the Crutcher relation which corresponds to a Bayesian model of
the maximum magnetic field strength versus density obtained from Zeeman
measurements (Crutcher et al., 2010).

$B=\begin{cases}B_{0}&n<n_{0}\\\
B_{0}\left(\frac{n}{n_{0}}\right)^{k}&n>n_{0},\end{cases}$ (3)

where $n_{\scriptscriptstyle 0}\sim 300$ cm-3, $B_{\scriptscriptstyle 0}\sim
10$ $\mu$G and $\kappa=2/3$ are estimated from Zeeman measurements of cloud
structures (Crutcher, 2012). We additionally plot the Davis-Chandrasekhar-
Fermi or DCF observations (method - Davis 1951; Chandrasekhar & Fermi 1953;
data - Pattle et al. 2023 and references therein) and Zeeman measurements
(Crutcher et al., 2010). The DCF method finds
$B_{\scriptscriptstyle\textrm{POS}}$, i.e. the position on the sky magnetic
field and the Zeeman measurements calculate
$B_{\scriptscriptstyle\textrm{LOS}}$, which is the line of sight magnetic
field. The DCF method estimates $B_{\scriptscriptstyle\textrm{POS}}$ through
the dispersion in polarisation angle from dust emission or extinction. This
can be used to estimate the disruption of the magnetic field by non-thermal
gas motions, which can be a measure of the Alfvén Mac number
$\mathcal{M}_{\scriptscriptstyle A}$.

Figure 7: Magnetic field - density relation for Arm1, Arm5 and Arm10 (left panel) as well as Cloud5 and Cloud50 (right panel) in the top row. Alfvén Mach number against density for those same models in the bottom row. These are calculated at 6 Myrs for the Arm models and 5 Myrs for the cloud models. DCF and Zeeman measurements of $B_{\scriptscriptstyle\textrm{POS}}$ and $B_{\scriptscriptstyle\textrm{LOS}}$ are shown in the top panel (Crutcher, 2012; Pattle et al., 2023). In the bottom panel only DCF measurements are shown, which correspond to $\mathcal{M}_{\scriptscriptstyle A}=\sigma_{\scriptscriptstyle\theta}/Q$ (see Pattle et al. 2021 for details). The observations correspond to zoomed in regions from Figure 2a and 2e of Pattle et al. (2023). Figure 8: The Alfvén Mach number, $\mathcal{M}_{\scriptscriptstyle A}$, is shown in the top panel and the mass to flux ratio, $\mu_{\scriptscriptstyle B}$, in the bottom panel for clouds from models Arm5 (blue) and Arm5∗ (orange). Arm5∗ is the same as Arm5, except this model does not include stellar feedback. All properties are plotted with respect to number density $n_{\scriptscriptstyle H}$. Table 3: Table of cluster masses for the Cloud models. The clusters are identified using HDBSCAN. Model | Cluster 1 (M⊙) | Cluster 2 (M⊙) | Cluster 3 (M⊙) | Cluster 4 (M⊙) | Cluster 5 (M⊙) | Cluster 6 (M⊙)
---|---|---|---|---|---|---
Cloud0 | 6704 | 2621 | 866 | 426 | 348 | 145
Cloud5 | 6735 | 2494 | 848 | 628 | 574 | 518
Cloud50 | 3651 | 1420 | 912 | 852 | - | -

The Arm and Cloud models roughly span the density range $n_{\scriptscriptstyle
H}\sim 10^{2}-10^{5}$ cm-3 and show better agreement with the Zeeman
measurements compared to the DCF measurements at these densities. The Arm
models find $B$ is fairly constant with density. For the Arm10 model, $B$
remains effectively constant across all densities, though for the Arm1 and
Arm5 models, $B$ starts to increase slightly with density at around
$n_{\scriptscriptstyle H}\gtrsim 3\times 10^{3}$ cm-3. The Cloud5 model shows
an increase in $B$ with density, which by eye matches the data well. We fit an
exponent of $\kappa=0.861$ to the simulated data. For the Cloud50 model
however, the magnetic field is fairly constant and does not match the
observations that well. Overall the higher magnetic field strength in the
Arm10 and Cloud50 models seems to impede the formation of dense, strongly
magnetic clouds and clumps.

###  3.4 Turbulence and magnetic fields

Combining the Alfvén speed, $\nu_{\scriptscriptstyle A}$ of a structure along
with the non thermal velocity dispersion, $\sigma_{\scriptscriptstyle v}$ of
the gas gives us the Alfvén Mach number, $\mathcal{M}_{A}=\sigma_{v}/v_{A}$.
This indicates the level of magnetic turbulence within a structure,
$\mathcal{M}_{\scriptscriptstyle A}<1$ is sub-Alfvénic and
$\mathcal{M}_{\scriptscriptstyle A}>1$ is super-Alfvénic turbulence. In our
simulations we compute $\sigma_{v}$ and $\nu_{\scriptscriptstyle A}$ directly
from the gas particles within a given structure.

In the bottom panel of Figure 7 we show the Alfvén Mach numbers for the Arm
and Cloud models compared to DCF measurements (described in §3.3). Pattle et
al. (2023) explains that the DCF measurements of
$\mathcal{M}_{\scriptscriptstyle A}$ assume sub-Alfvénic turbulence within the
gas, which is why these observations generally find
$\mathcal{M}_{\scriptscriptstyle A}<1$. Zeeman measurements span lower
magnetic fields which would correspond to higher Alfvén Mach numbers more in
line with our models. Sub-Alfvénic turbulence is common in our Arm and Cloud
models but we also observe significant super-Alfvénic turbulence within clouds
and clumps. The mean Mach numbers within the Arm models Arm1, Arm5 and Arm10
are $\mathcal{M}_{\scriptscriptstyle A}=$ 14.0, 4.93 and 2.05 respectively,
i.e. super-Alfvénic. The median values for the Arm models are super-Alfvénic
for Arm1 and roughly trans-Alfvénic for Arm5 and Arm10. Our cloud models
reflect this behaviour with mean values super-Alfvénic and median values
trans-Alfvénic, but Cloud5 shows more sub-Alfvénic clumps compared to Cloud50.

We investigate the relationship between feedback and magnetic fields by
calculating the Alfvén Mach number and criticality of cloud structures within
Arm5 and a new model Arm5∗. This new model has the same initial conditions as
Arm5 but evolves without stellar feedback. In the top panel of Figure 8 we
show the Alfvén Mach Number $\mathcal{M}_{\scriptscriptstyle A}$. We find
significantly more super-Alfvénic clouds, as would be expected for feedback
driving turbulence, in Arm5 compared to Arm5∗, as well as more higher density
clouds. The critical parameter for clouds in both models is shown in the
bottom panel. For structures of the same density we find the criticality of
clouds to be roughly the same in Arm5 and Arm5∗, indicating that feedback is
not obviously effecting magnetic criticality.

Figure 9: (a) - The magnetic critical parameter $\mu_{\scriptscriptstyle B}$
of clouds in Arm5 is plotted, computed using using $B_{\scriptscriptstyle x}$,
$B_{\scriptscriptstyle y}$ or $B_{\scriptscriptstyle z}$ only compared to 3D
$B_{\scriptscriptstyle xyz}$. (b) - $\mu_{\scriptscriptstyle B}$ calculated
using $B_{\scriptscriptstyle\textrm{LOS(x)}}$ or
$B_{\scriptscriptstyle\textrm{LOS(z)}}$ compared to
$B_{\scriptscriptstyle\textrm{LOS(y)}}$ (see text). These measures of
$\mu_{\scriptscriptstyle B}$ are calculated using the column density along the
line rather than the flux, but give similar values to the flux method.

###  3.5 Clusters and magnetic fields

For the Cloud models, the resolution is sufficiently high that we are able to
identify clusters from the sink particles formed in the simulation. We use
HDBSCAN (Campello et al., 2013) to identify clusters, since this is used
commonly by observers (Joncour et al., 2020). We also used this in previous
work (Ali et al., 2023). We set the minimum cluster size min_cluster_size =
15, which sets the minimum number of particles in a cluster. We define
min_samples = 3, which is a measure of how conservative the clustering
algorithm is, and how many points remain as noise.

From Figure 1, we see by eye that the clusters seem less extensive in Cloud50,
whereas for Cloud0 and Cloud5 the distribution of sink particles looks
similar. We show the number of clusters, and the cluster masses, from the
clusters identified with HDBSCAN in Table 3. The number of clusters is the
same for Cloud0 and Cloud5, and the masses of the clusters fairly similar
(though the masses of the smallest clusters tend to be a little higher in
Cloud5). By contrast, for Cloud50, there are only 4 clusters, instead of 6.
The masses of the two most massive clusters are reduced by about 40%.

##  4 Discussion

###  4.1 Directional dependence of $\mu_{\scriptscriptstyle B}$

Our simulations provide us with the full 3D magnetic field properties. DCF
methods only recover 2D magnetic field information, i.e. the position on the
sky magnetic field $B_{\scriptscriptstyle\textrm{POS}}$. The most direct
method, Zeeman splitting, measures the magnetic field strength
$B_{\scriptscriptstyle\textrm{LOS}}$ in the direction of the line of sight
only. We consider how the magnetic critical parameter depends on $B$, assuming
that we only have the magnetic field information in a particular direction,
and using a line of sight method.

We first see how magnetic field direction affects $\mu_{\scriptscriptstyle
B}$, as calculated in equation 1. For simplicity, we take $B_{x}$, $B_{y}$,
$B_{z}$, for equation 1, noting that for our region, the magnetic field is
most aligned with the $y$ direction. In panel (a) of Figure 9 we show the
critical mass to flux ratio for cloud structures in Arm5, computed with the
full 3D field strength $B_{\scriptscriptstyle\textrm{xyz}}$ and single field
directions $B_{\scriptscriptstyle\textrm{x}}$,
$B_{\scriptscriptstyle\textrm{y}}$ or $B_{\scriptscriptstyle\textrm{z}}$. We
observe that $\mu_{\scriptscriptstyle B_{\scriptscriptstyle\textrm{y}}}$
produces a similar mass to flux ratio to $\mu_{\scriptscriptstyle
B_{\scriptscriptstyle\textrm{xyz}}}$ whereas $\mu_{\scriptscriptstyle
B_{\scriptscriptstyle\textrm{x}}}$ and $\mu_{\scriptscriptstyle
B_{\scriptscriptstyle\textrm{z}}}$ produce significantly larger critical
parameters, i.e. more supercritical. We find $<\mu_{B_{\scriptscriptstyle
xyz}}/\mu_{B_{\scriptscriptstyle x}}>=3.7$ and $<\mu_{B_{\scriptscriptstyle
xyz}}/\mu_{B_{\scriptscriptstyle z}}>=4.3$ corresponding to the peaks of the
histogram in the top panel of Figure 9.

The higher $\mu_{\scriptscriptstyle B_{\textrm{x}}}$, $\mu_{\scriptscriptstyle
B_{\textrm{z}}}$ are due to lower magnetic field strengths in these
directions, compared to the $y$ direction. This is because the global field is
most aligned with the $y$ direction. We can estimate the predicted directional
dependence as follows. Observations find that the random component of the
field is about 1/3 of the ordered component (Fletcher, 2010). If we are
looking in the direction of the ordered component, which will tend to be the
rotation of the galaxy or along the spiral arms, we will see a maximum
difference of a factor of 3 between the magnetic field strength, or critical
parameter, compared to perpendicular directions. If we look in the plane of
the disc but perpendicular to the ordered field, we would expect minimal
directional dependence. But overall we would expect direction to contribute a
factor of 3 uncertainty.

In panel (b) we computed the critical parameters using a column density
method, again for the Arm5 model. We take lines of sight over a grid (with
spacing 1 pc), and for any line of sight which passes within a smoothing
length of particles in the cloud, we compute the column density along the line
of sight from material in the cloud, and the mean magnetic field over those
constituent particles. The magnetic criticality is then calculated as

$\mu_{B_{\scriptscriptstyle\textrm{LOS}}}=\frac{\mu_{\scriptscriptstyle
H}N}{B},$ (4)

where $\mu_{\scriptscriptstyle H}$ is the mean molecular weight and $N$ is the
column density (Crutcher, 1999). This is more analogous to Zeeman
observations. We compare the LOS derived critical parameters in the y
direction to the other weaker field strength components. This gives
$<\mu_{B_{\scriptscriptstyle\textrm{LOS(y)}}}/\mu_{B_{\scriptscriptstyle\textrm{LOS(x)}}}>=3.1$
and
$<\mu_{B_{\scriptscriptstyle\textrm{LOS(y)}}}/\mu_{B_{\scriptscriptstyle\textrm{LOS(z)}}}>=4.1$.
So both the values for $\mu$, and the relative differences between directions,
are similar for both types of calculations of $\mu$.

###  4.2 Magnetic fields

We have run models with varying magnetic field strengths. Morphologically, the
subcritical Cloud50 model shows a large difference in the structure of the
gas, and the reduced effect of feedback, compared with the purely
hydrodynamical model. The magnetic field also has a much greater impact on
star formation, reducing the amount of star formation by as much as 50%, and
the mass of clusters by a factor of around a half. For Cloud5, the morphology
of the gas is more similar to the hydrodynamical model, the clusters are
similar and the star formation rate reduced by a smaller amount. So in this
case, the behaviour of Cloud5 seems more representative of the weak field
scenario, whilst Cloud50 is more representative of the strong field scenario.

The main diagnostic for comparing with observations we have shown is the
B-$\rho$ relation. We see that for the strongest field case, Cloud50, the
results do not match the observations particularly well, in that $B$ is flat
versus density instead of increasing, whereas Cloud5 matches much better. We
further tested this by running a model Arm50 (with a 50 $\mu$G initial field),
and this model also produced points that disagreed with the observations,
showing higher $B$ values which are flat with density and cross the Crutcher
relation. For both of these models with 50 $\mu$G fields, the magnetic fields
are strongly preventing gravitational collapse. As discussed in Crutcher et
al. (2010), the transition from a flat relation to a $B\propto\rho^{2/3}$
relation is expected to occur when clouds can contract via gravity. Thus it is
not surprising that these stronger field models show no upturn in $B$. The
Arm50 model also looked morphologically dissimilar to what we would expect,
with filaments forming perpendicular, rather than parallel to the arm
(extending the trend we see in Figure 4). For larger, lower density, arm
scales we would expect to see filaments more parallel to the magnetic field,
and both roughly aligned with the spiral arms, like Arm1 and a lesser extent
Arm5. So overall, our models with weaker fields are more consistent with
observations. A caveat to our models is that we do not include ambipolar
diffusion, which could allow gravitational collapse even in the strong field
cases, although for our Arm models at least, we would not expect ambipolar
diffusion to be relevant on these scales.

Observationally, we see from the $B-\rho$ relation that the DCF measurements
tend to be higher than the Zeeman measurements. The DCF method is more
uncertain compared to the Zeeman method, since the latter measures the field
directly rather than inferring the field from the dispersion of polarisation
vectors. Our models show magnetic field strengths which agree better with the
Zeeman, rather than the DCF measurements, and thus we find that the Zeeman
measurements are a better match to the models. As mentioned above, for higher
magnetic fields, our models still don’t fit the DCF measurements, because the
B-$\rho$ relation is too flat.

##  5 Conclusions

We have presented 3D smooth particle magnetohydrodynamic simulations of star
formation on kpc and pc scales, following the evolution of a section of spiral
arm and a single cloud structure. Listed below is a summary of our main
conclusions from the results of our simulations.

  * (1)

We find that magnetic fields cause smoothing of dense structures leading to
smoother feedback shells and filamentary networks compared to our purely
hydrodynamic models.

  * (2)

Magnetic fields suppress star formation rates on both kpc and pc scales. The
strongest suppression of the SFR on kpc scales occurs in the highest strength
model (Arm10), where the SFR is reduced by roughly 30%, comparable to previous
results (e.g. Wibking & Krumholz 2023). On pc scales our strong field model
(with $B_{\scriptscriptstyle 0}=50\leavevmode\nobreak\ $ $\mu$G) shows
suppression of up to $\sim$50%. The decrease in star formation leads to a
reduction in the number of clusters from 6 to 4, and decrease in mass of the
most massive cluster by 40%, in the strongest field model. However, with a
more moderate field ($B_{\scriptscriptstyle 0}=5\leavevmode\nobreak\ $
$\mu$G), the number and masses of clusters are similar to the hydrodynamic
case.

  * (3)

With less star formation and therefore feedback, there is also less gas
dispersal in our Cloud model with the strongest field. This likely prolongs
the cloud’s lifetime and duration of star formation compared to our
hydrodynamical model.

  * (4)

Cloud structures identified in the Arm models tend to be more filament like
and aligned parallel to their magnetic fields in low magnetic field strength
models and perpendicularly in higher magnetic field strength models. The
transition in orientations at higher magnetic field strengths is consistent
with previous work (Soler & Hennebelle, 2017; Seifried et al., 2020; Dobbs &
Wurster, 2021; Barreto-Mota et al., 2021; Mazzei et al., 2023). We do not find
a transition between the flow and magnetic field direction, both Arm1 and
Arm10 show flow aligned to the direction of the magnetic field. Filamentary
structures within our cloud models do not show a clear alignment preference,
contrary to similar scale structures observed inside molecular clouds (e.g.
Planck Collaboration et al. 2016; Soler et al. 2017; Alina et al. 2019; Lee et
al. 2021).

  * (5)

We see a large spread in criticality for both clumps and clouds throughout the
simulations. We surprisingly see more supercritical low mass clumps at later
times, particularly in Cloud50. We attribute these to feedback sweeping gas
into dense shells, with small clumps occurring as the shell fragments.

  * (6)

We find the critical parameter has a directional dependence. We compared
calculations of the critical parameter using only one component of the
magnetic field, $B_{\scriptscriptstyle x}$, $B_{\scriptscriptstyle y}$ and
$B_{\scriptscriptstyle z}$, giving us $\mu_{\scriptscriptstyle
B_{\scriptscriptstyle x}}$, $\mu_{\scriptscriptstyle B_{\scriptscriptstyle
y}}$ and $\mu_{\scriptscriptstyle B_{\scriptscriptstyle z}}$. This was
compared to $\mu_{\scriptscriptstyle B_{\scriptscriptstyle xyz}}$ where
$B_{\scriptscriptstyle xyz}$ is the total 3D magnetic field strength. We find
$\mu_{\scriptscriptstyle B_{\scriptscriptstyle x}}$ and
$\mu_{\scriptscriptstyle B_{\scriptscriptstyle z}}$ overestimate the critical
parameter by roughly 3-4 times compared to $\mu_{\scriptscriptstyle
B_{\scriptscriptstyle xyz}}$, whereas the component $\mu_{\scriptscriptstyle
B_{\scriptscriptstyle y}}$ produces effectively the same values as
$\mu_{\scriptscriptstyle B_{\scriptscriptstyle xyz}}$. This level of variation
might be expected if the random component of the galactic magnetic field is of
order 1/3 the ordered component. We also compared calculating the critical
parameter using the line of sight column density versus using the flux and
found the methods produced similar results.

  * (7)

We determine the $B-\rho$ relation for our models, finding that the magnetic
field measurements for our clumps and clouds agree better with Zeeman
observations compared to DCF measurements. Our stronger field models
($B_{\scriptscriptstyle 0}=50\leavevmode\nobreak\ $ $\mu$G and 10 $\mu$G) show
a flat $B-\rho$ relation, whereas Cloud5 ($B_{\scriptscriptstyle
0}=5\leavevmode\nobreak\ $ $\mu$G) and to a lesser extend Arm1 and Arm5 show
an increasing $B-\rho$ relationship at higher densities. The change from flat
to increasing $B-\rho$ is expected to coincide with gravitational contraction
(Crutcher et al., 2010), which may explain why our stronger field models,
where star formation is significantly impeded, exhibit a flat $B-\rho$
relationship.

  * (8)

DCF measurements predict larger magnetic fields and so greater Alfvén speeds,
recovering on average sub-Alfvénic ($\mathcal{M}_{\scriptscriptstyle A}$)
turbulence on density scales $n_{\scriptscriptstyle H}\leq 10^{6}$ cm-3. Our
models show significant numbers of super-Alfvénic structures on both arm and
cloud scales for densities $n_{\scriptscriptstyle H}\sim 10^{2}-10^{5}$.

  * (9)

Finally we report that feedback does not effect the magnetic criticality of
clouds. Instead we observe more super-Alfvénic clouds in magnetic models with
feedback compared to magnetic models without feedback.

As discussed previously, changing the resolution does not seem to affect the
magnetic properties, but there are a few further caveats. One is that we start
with a uniform field rather than taking a field from a global galaxy
simulation. As we don’t have a suitable MHD galaxy simulation, we cannot test
directly what difference this makes. However in Appendix C we consider the
magnetic flux ratios over the course of the Arm simulations, and show they are
not hugely different at later times, the main change is that the spread in the
flux ratios increases. A further caveat is that when we increase the
resolution we do not follow turbulence on the smallest scales, which may
influence when and where star formation occurs. We tested this using a low
resolution simulation equivalent to Arm5LR, but where we inject turbulence on
small scales. We do this by adding a velocity dispersion when we split the
particles, by sampling from a Gaussian distribution of width equal to the
velocity dispersion expected from an empirical Larson relation. We find little
difference in morphology and number of sinks formed compared to Arm5LR.

## Acknowledgements

We would like to acknowledge James Wurster for his contribution to this work,
in particular advice regarding using a boundary medium. We additionally
acknowledge Kate Pattle for sharing the DCF data which enabled us to compare
our results to observations. As well as this we would like to acknowledge the
referee for their helpful comments and suggestions. Lastly we acknowledge Tim
Naylor for our useful discussions. All authors are funded by the European
Research Council H2020-EU.1.1 titled the ICYBOB project, grant number 818940.
Our simulations were performed with the DiRAC DIaL system, facilitated by the
University of Leicester IT Services, forming part of the STFC DiRAC HPC
Facility (www.dirac.ac.uk). The equipment was funded by BEIS capital funding
via STFC capital grants ST/K000373/1 and ST/R002363/1 and STFC DiRAC
Operations grant ST/K001014/1. DiRAC is part of the National E-Infrastructure.

## Data availability statement

Upon reasonable request to the author, all data in this work will be shared.

## References

  * Ali et al. (2022) Ali A. A., Bending T. J. R., Dobbs C. L., 2022, Monthly Notices of the Royal Astronomical Society, 510, 5592
  * Ali et al. (2023) Ali A. A., Dobbs C. L., Bending T. J. R., Buckner A. S. M., Pettitt A. R., 2023, MNRAS, 524, 555
  * Alina et al. (2019) Alina D., Ristorcelli I., Montier L., Abdikamalov E., Juvela M., Ferrière K., Bernard J. P., Micelotta E. R., 2019, MNRAS, 485, 2825
  * Auddy et al. (2022) Auddy S., Basu S., Kudoh T., 2022, ApJ, 928, L2
  * Barreto-Mota et al. (2021) Barreto-Mota L., de Gouveia Dal Pino E. M., Burkhart B., Melioli C., Santos-Lima R., Kadowaki L. H. S., 2021, MNRAS, 503, 5425
  * Bate et al. (1995a) Bate M. R., Bonnell I. A., Price N. M., 1995a, MNRAS, 277, 362
  * Bate et al. (1995b) Bate M. R., Bonnell I. A., Price N. M., 1995b, MNRAS, 277, 362
  * Bate et al. (1995c) Bate M. R., Bonnell I. A., Price N. M., 1995c, MNRAS, 277, 362
  * Beck (2015) Beck R., 2015, A&ARv, 24, 4
  * Bending et al. (2020) Bending T. J. R., Dobbs C. L., Bate M. R., 2020, MNRAS, 495, 1672
  * Benz (1990) Benz W., 1990, in Buchler J. R., ed., Numerical Modelling of Nonlinear Stellar Pulsations Problems and Prospects. p. 269
  * Benz et al. (1990) Benz W., Bowers R. L., Cameron A. G. W., Press W. H. ., 1990, ApJ, 348, 647
  * Børve et al. (2001) Børve S., Omang M., Trulsen J., 2001, ApJ, 561, 82
  * Campello et al. (2013) Campello R. J. G. B., Moulavi D., Sander J., 2013, in Pei J., Tseng V. S., Cao L., Motoda H., Xu G., eds, Advances in Knowledge Discovery and Data Mining. Springer Berlin Heidelberg, Berlin, Heidelberg, pp 160–172
  * Chandrasekhar & Fermi (1953) Chandrasekhar S., Fermi E., 1953, ApJ, 118, 116
  * Chen et al. (2020) Chen C.-Y., et al., 2020, MNRAS, 494, 1971
  * Cox (2005) Cox D. P., 2005, ARA&A, 43, 337
  * Crutcher (1999) Crutcher R. M., 1999, ApJ, 520, 706
  * Crutcher (2004) Crutcher R. M., 2004, in Uyaniker B., Reich W., Wielebinski R., eds, The Magnetized Interstellar Medium. pp 123–132
  * Crutcher (2012) Crutcher R. M., 2012, ARA&A, 50, 29
  * Crutcher et al. (2010) Crutcher R. M., Wandelt B., Heiles C., Falgarone E., Troland T. H., 2010, ApJ, 725, 466
  * Davis (1951) Davis L., 1951, Physical Review, 81, 890
  * Dedner et al. (2002) Dedner A., Kemm F., Kröner D., Munz C. D., Schnitzer T., Wesenberg M., 2002, Journal of Computational Physics, 175, 645
  * Dobbs & Price (2008) Dobbs C. L., Price D. J., 2008, MNRAS, 383, 497
  * Dobbs & Pringle (2013) Dobbs C. L., Pringle J. E., 2013, MNRAS, 432, 653
  * Dobbs & Wurster (2021) Dobbs C. L., Wurster J., 2021, MNRAS, 502, 2285
  * Dobbs et al. (2011) Dobbs C. L., Burkert A., Pringle J. E., 2011, MNRAS, 417, 1318
  * Dobbs et al. (2017) Dobbs C. L., et al., 2017, MNRAS, 464, 3580
  * Dobbs et al. (2022a) Dobbs C. L., Bending T. J. R., Pettitt A. R., Bate M. R., 2022a, MNRAS, 509, 954
  * Dobbs et al. (2022b) Dobbs C. L., Bending T. J. R., Pettitt A. R., Buckner A. S. M., Bate M. R., 2022b, MNRAS, 517, 675
  * Federrath (2015) Federrath C., 2015, MNRAS, 450, 4035
  * Federrath & Klessen (2012) Federrath C., Klessen R. S., 2012, ApJ, 761, 156
  * Fletcher (2010) Fletcher A., 2010, in Kothes R., Landecker T. L., Willis A. G., eds, Astronomical Society of the Pacific Conference Series Vol. 438, The Dynamic Interstellar Medium: A Celebration of the Canadian Galactic Plane Survey. p. 197 (arXiv:1104.2427), doi:10.48550/arXiv.1104.2427
  * Ganguly et al. (2023) Ganguly S., Walch S., Seifried D., Clarke S. D., Weis M., 2023, MNRAS, 525, 721
  * Ganguly et al. (2024) Ganguly S., Walch S., Clarke S. D., Seifried D., 2024, MNRAS, 528, 3630
  * Girichidis (2021) Girichidis P., 2021, MNRAS, 507, 5641
  * Glover & Mac Low (2007) Glover S. C. O., Mac Low M.-M., 2007, ApJS, 169, 239
  * Heiles & Troland (2005) Heiles C., Troland T. H., 2005, ApJ, 624, 773
  * Hennebelle & Iffrig (2014) Hennebelle P., Iffrig O., 2014, A&A, 570, A81
  * Herrington et al. (2023) Herrington N. P., Dobbs C. L., Bending T. J. R., 2023, MNRAS, 521, 5712
  * Hu et al. (2023) Hu Z., Wibking B. D., Krumholz M. R., 2023, MNRAS, 521, 5604
  * Inoue & Inutsuka (2009) Inoue T., Inutsuka S.-i., 2009, ApJ, 704, 161
  * Inoue & Inutsuka (2016) Inoue T., Inutsuka S.-i., 2016, ApJ, 833, 10
  * Joncour et al. (2020) Joncour I., Buckner A., Khalaj P., Moraux E., Motte F., 2020, arXiv e-prints, p. arXiv:2006.07830
  * Klassen et al. (2017) Klassen M., Pudritz R. E., Kirk H., 2017, MNRAS, 465, 2254
  * Körtgen & Banerjee (2015) Körtgen B., Banerjee R., 2015, MNRAS, 451, 3340
  * Körtgen et al. (2018) Körtgen B., Banerjee R., Pudritz R. E., Schmidt W., 2018, MNRAS, 479, L40
  * Krumholz (2011) Krumholz M. R., 2011, in Telles E., Dupke R., Lazzaro D., eds, American Institute of Physics Conference Series Vol. 1386, XV Special Courses at the National Observatory of Rio de Janeiro. pp 9–57 (arXiv:1101.5172), doi:10.1063/1.3636038
  * Lee et al. (2021) Lee D., et al., 2021, ApJ, 918, 39
  * Mac Low & Klessen (2004) Mac Low M.-M., Klessen R. S., 2004, Reviews of Modern Physics, 76, 125
  * Mazzei et al. (2023) Mazzei R., Li Z.-Y., Chen C.-Y., Fissel L., Chen M., Park J., 2023, MNRAS, 521, 3830
  * McClure-Griffiths et al. (2006) McClure-Griffiths N. M., Dickey J. M., Gaensler B. M., Green A. J., Haverkorn M., 2006, ApJ, 652, 1339
  * Mestel (1966) Mestel L., 1966, MNRAS, 133, 265
  * Mestel & Paris (1984) Mestel L., Paris R. B., 1984, A&A, 136, 98
  * Morris & Monaghan (1997) Morris J. P., Monaghan J. J., 1997, Journal of Computational Physics, 136, 41
  * Mouschovias (1979) Mouschovias T. C., 1979, ApJ, 228, 475
  * Mouschovias (1981) Mouschovias T. C., 1981, in Sugimoto D., Lamb D. Q., Schramm D. N., eds, Vol. 93, Fundamental Problems in the Theory of Stellar Evolution. pp 27–59
  * Mouschovias & Ciolek (1999) Mouschovias T. C., Ciolek G. E., 1999, in Lada C. J., Kylafis N. D., eds, NATO Advanced Study Institute (ASI) Series C Vol. 540, The Origin of Stars and Planetary Systems. p. 305
  * Mouschovias & Spitzer (1976) Mouschovias T. C., Spitzer L. J., 1976, ApJ, 210, 326
  * Nakano (1978) Nakano T., 1978, PASJ, 30, 681
  * Padoan & Nordlund (1999) Padoan P., Nordlund Å., 1999, ApJ, 526, 279
  * Padoan & Nordlund (2011) Padoan P., Nordlund Å., 2011, ApJ, 730, 40
  * Pakmor & Springel (2013) Pakmor R., Springel V., 2013, MNRAS, 432, 176
  * Parker (1967) Parker E. N., 1967, ApJ, 149, 535
  * Pattle et al. (2021) Pattle K., et al., 2021, ApJ, 907, 88
  * Pattle et al. (2023) Pattle K., Fissel L., Tahani M., Liu T., Ntormousi E., 2023, in Inutsuka S., Aikawa Y., Muto T., Tomida K., Tamura M., eds, Astronomical Society of the Pacific Conference Series Vol. 534, Protostars and Planets VII. p. 193 (arXiv:2203.11179), doi:10.48550/arXiv.2203.11179
  * Peters et al. (2011) Peters T., Klessen R. S., Mac Low M. M., Banerjee R., 2011, in Stellar Clusters & Associations: A RIA Workshop on Gaia. pp 229–234 (arXiv:1110.2892), doi:10.48550/arXiv.1110.2892
  * Pettitt et al. (2015) Pettitt A. R., Dobbs C. L., Acreman D. M., Bate M. R., 2015, MNRAS, 449, 3911
  * Planck Collaboration et al. (2016) Planck Collaboration et al., 2016, A&A, 586, A135
  * Portegies Zwart & Verbunt (2012) Portegies Zwart S. F., Verbunt F., 2012, SeBa: Stellar and binary evolution, Astrophysics Source Code Library, record ascl:1201.003 (ascl:1201.003)
  * Price (2012) Price D. J., 2012, Journal of Computational Physics, 231, 759
  * Price & Bate (2008) Price D. J., Bate M. R., 2008, MNRAS, 385, 1820
  * Price & Bate (2009) Price D. J., Bate M. R., 2009, MNRAS, 398, 33
  * Price & Monaghan (2007) Price D. J., Monaghan J. J., 2007, MNRAS, 374, 1347
  * Seifried et al. (2020) Seifried D., Walch S., Weis M., Reissl S., Soler J. D., Klessen R. S., Joshi P. R., 2020, MNRAS, 497, 4196
  * Sharma et al. (2022) Sharma E., Gopinathan M., Soam A., Lee C. W., Seshadri T. R., 2022, MNRAS, 517, 1138
  * Shu (1983) Shu F. H., 1983, ApJ, 273, 202
  * Shu et al. (1987) Shu F. H., Adams F. C., Lizano S., 1987, ARA&A, 25, 23
  * Soler & Hennebelle (2017) Soler J. D., Hennebelle P., 2017, A&A, 607, A2
  * Soler et al. (2013) Soler J. D., Hennebelle P., Martin P. G., Miville-Deschênes M. A., Netterfield C. B., Fissel L. M., 2013, ApJ, 774, 128
  * Soler et al. (2017) Soler J. D., et al., 2017, A&A, 603, A64
  * Van Loo et al. (2015) Van Loo S., Tan J. C., Falle S. A. E. G., 2015, ApJ, 800, L11
  * Vázquez-Semadeni et al. (2011) Vázquez-Semadeni E., Banerjee R., Gómez G. C., Hennebelle P., Duffin D., Klessen R. S., 2011, MNRAS, 414, 2511
  * Wang & Abel (2009) Wang P., Abel T., 2009, ApJ, 696, 96
  * Wibking & Krumholz (2023) Wibking B. D., Krumholz M. R., 2023, MNRAS, 521, 5972
  * Wurster et al. (2014) Wurster J., Price D., Ayliffe B., 2014, MNRAS, 444, 1104
  * Wurster et al. (2016) Wurster J., Price D. J., Bate M. R., 2016, MNRAS, 457, 1037
  * Wurster et al. (2021) Wurster J., Bate M. R., Bonnell I. A., 2021, MNRAS, 507, 2354
  * Xu et al. (2019) Xu S., Ji S., Lazarian A., 2019, ApJ, 878, 157
  * Zamora-Avilés et al. (2018) Zamora-Avilés M., Vázquez-Semadeni E., Körtgen B., Banerjee R., Hartmann L., 2018, MNRAS, 474, 4824

##  Appendix A Resolution test

We produced a new model named Arm5LR which has the same initial conditions as
Arm5 but has a lower resolution of 10.5 M⊙ per particle. Figure 10 shows a
comparison of magnetic properties of clouds identified in both Arm5 and
Arm5LR. We find that all magnetic properties are similar between Arm5 and the
lower resolution model Arm5LR.

Figure 10: Histograms of mass to flux ratio $\mu_{B}$, the ratio of Alfvén
speed to sound speed $\nu_{A}/\nu_{s}$ and the ratio of magnetic to kinetic
energy of identified cloud structures in Arm5 (orange line) and new model
Arm5LR (blue dashed line), calculated at 4 Myrs.

##  Appendix B Classifying structures

Figure 11: Structures classified as filaments, sheets, curved sheets and
spheroidal. Circle points are particles that make up these structures and the
cross is the centre of mass of the structure. These are examples of cloud
structures from our Arm10 model.

We present examples of the classified structures mentioned in section 3.2.
Figure 11 shows cloud structures that correspond to filaments, sheets, curved
sheets and spheroids.

Figure 12: Mass to flux ratios $\mu_{B}$ of clouds in Arm1 (green), Arm5
(orange) and Arm10 (blue) at times t = 0, 2 and 5 Myrs.

##  Appendix C initial distribution of magnetic properties

The initial conditions of the Arm and Cloud models are extracted from non-MHD
galaxy models. For our Arm and Cloud models the magnetic field in our initial
conditions are generated by setting up a toroidal magnetic field. We compare
the mass to flux ratio of clouds identified at t = 0, so from the initial
conditions, to the rough middle and end of the simulation (4 and 6 Myrs
respectively). These are shown in Figure 12, and we find the distributions of
mass to flux ratios exhibit similar average values from t = 0 to t = 6 Myrs,
although the spread in the values increases with time.